transformer vibration and its application to condition monitoring

Transformer vibration and its application to

condition monitoring

Yuxing Wang

B. Eng., M. Eng.

This thesis is presented for the

Degree of Doctor of Philosophy

at the University of Western Australia

School of Mechanical and Chemical Engineering

April 2015

i

Abstract

The electrical power is an important part of daily life and a necessity for the

development of modern industry. The dependency of a country’s economic

development on electrical power is growing rapidly. Consequently, planning, designing,

constructing, and maintaining power delivery systems must keep pace with the

escalating demand of such development. Power transformers are a key component of a

power transmission system, and condition monitoring and failure diagnosis techniques

are commonly required by transformer owners for reliability and maintenance purposes.

Despite several decades of research into transformer vibration and condition

monitoring techniques, state-of-the-art development in this area still falls short in the

understanding of the mechanisms involved and in industry implementation. The

objective of this thesis therefore is to investigate the vibration characteristics of a power

transformer with and without structural damage and to develop a vibration-based

transformer condition monitoring technique. It is hoped that this work could give a

better understanding of transformer vibration and its application to condition monitoring.

To that end, several aspects of transformer vibration are studied experimentally

and numerically, including its excitation forces, modal characteristics, and vibration

frequency responses. The finite element (FE) method is employed as the main approach

for numerical analysis of the aforementioned aspects. The effect of the arrangement of

ferromagnetic parts on the modelling of winding electromagnetic (EM) forces is

discussed in detail with the purpose of improving its modelling accuracy. Special

considerations, i.e., the anisotropic mechanical properties of core lamination, of

transformer vibration modelling are summarised based on the traditional experimental

modal analysis. Vibration features of a transformer with structural anomalies, especially

ii

with cases of winding failure, are investigated using a verified FE model. In addition,

the frequency response function and its variations caused by structural anomalies are

studied experimentally under both mechanical and electrical excitations.

It is shown that a structural anomaly will produce shifts in the natural frequency

and changes in the vibration response. The experimental results also demonstrate that

the transformer mechanical resonance can be excited by internal electrical excitations,

which enables operational modal analysis (OMA) and OMA-based online monitoring.

An algorithm based on the time-domain NExt/ITD method is employed as an OMA

technique to identify transformer modal parameters. The features of transformer

vibration and operational conditions are considered in the proposed algorithm, which

improves the identification accuracy in some cases. The identification method is also

applied to the same transformer with core and winding anomalies. Results show that the

OMA method is capable of identifying transformer modal parameters and thus can be

utilised for online condition monitoring.

iii

Content

Abstract ............................................................................................................................. i

Content ............................................................................................................................ iii

List of Figures ................................................................................................................. vi

List of Tables .................................................................................................................. xi

Acknowledgements ....................................................................................................... xiii

Declaration of Authorship ........................................................................................... xiv

Chapter 1 General Introduction .................................................................................... 1

1.1 Introduction ................................................................................................... 1

1.2 Thesis Focus .................................................................................................. 5

1.3 Thesis Organisation ....................................................................................... 6

Chapter 2 Accurate Modelling of Transformer Forces ............................................... 9

2.1 Introduction ................................................................................................... 9

2.2 Literature review ......................................................................................... 10

2.3 EM force calculation using DFS and FE methods ...................................... 14

2.3.1 General formulation of the DFS method ............................................ 14

2.3.2 Comparison of the DFS and FE methods ........................................... 19

2.3.3 Transformer EM force calculation on a 3D symmetric model ........... 23

2.4 Influential factors in modelling transformer EM forces .............................. 25

2.4.1 Shortcomings of the 2D model in EM force calculation ..................... 25

2.4.2 EM forces in the provision of magnetic flux shunts ............................ 30

2.5 Conclusion ................................................................................................... 35

Chapter 3 Modelling of Transformer Vibration ........................................................ 37

3.1 Introduction ................................................................................................. 37

3.2 Literature review ......................................................................................... 38

iv

3.3 Modelling setup and strategy based on FE method..................................... 43

3.3.1 Modelling considerations for the transformer core ............................ 44

3.3.2 Modelling considerations for the transformer winding ...................... 47

3.3.3 FE model of the test transformer ........................................................ 49

3.4 FE model verification by means of modal analysis .................................... 50

3.4.1 Modal test descriptions for a single-phase transformer ..................... 51

3.4.2 Modal analysis of the single-phase transformer ................................. 53

3.4.3 Numerical simulation of transformer frequency response ................. 65

3.5 Simulation of transformer vibration with winding damage ........................ 66

3.6 Conclusions ................................................................................................. 70

Chapter 4 Mechanically and Electrically Excited Vibration Frequency Response

Functions ........................................................................................................................ 72

4.1 Introduction ................................................................................................. 72

4.2 Methodology ............................................................................................... 74

4.3 Description of experiments ......................................................................... 76

4.4 Results and discussion ................................................................................. 78

4.4.1 FRF due to mechanical excitation ...................................................... 78

4.4.2 FRF due to electrical excitation ......................................................... 84

4.4.3 Effects of different clamping conditions ............................................. 89

4.4.4 FRFs of a 110 kV/50 MVA 3-phase power transformer ..................... 91

4.5 Conclusions ................................................................................................. 91

Chapter 5 Changes in the Vibration Response of a Transformer with Faults ........ 93

5.1 Introduction ................................................................................................. 93

5.2 Theoretical background ............................................................................... 96

5.3 Description of experiments ......................................................................... 97

v

5.4 Results and discussion ................................................................................. 99

5.4.1 Vibration response due to core looseness ......................................... 100

5.4.2 Vibration response due to winding looseness ................................... 103

5.4.3 Vibration response due to missing insulation spacers ...................... 106

5.4.4 Variation of the high-frequency vibration response ......................... 109

5.5 Conclusion ................................................................................................. 113

Chapter 6 Applications of Operational Modal Analysis to Transformer Condition

Monitoring ................................................................................................................... 116

6.1 Introduction ............................................................................................... 116

6.2 Theoretical background ............................................................................. 120

6.3 Feasibility analysis .................................................................................... 125

6.4 Operation verification ............................................................................... 129

6.4.1 OMA for a 10-kVA transformer ........................................................ 129

6.4.2 Structural damage detection based on transformer OMA ................ 133

6.5 Conclusion ................................................................................................. 135

Chapter 7 ..................................................................................................................... 137

Conclusions and Future Work ................................................................................... 137

7.1 Conclusions ............................................................................................... 137

7.2 Future prospects ........................................................................................ 142

Appendix A Further Discussion of Transformer Resonances and Vibration at

Harmonic Frequencies ................................................................................................ 144

Appendix B Voltage and Vibration Fluctuations in Power Transformers ............ 149

Nomenclature............................................................................................................... 171

References .................................................................................................................... 173

Publications originated from this thesis .................................................................... 184

vi

List of Figures

Figure 2.1. The 2D symmetric model of a 10-kVA small-distribution transformer. ...... 17

Figure 2.2. Comparison of leakage flux density (T) in the axial direction for the (a) DFS

and (b) FE results. ........................................................................................................... 21

Figure 2.3. Comparison of leakage flux density (T) in the radial direction for the (a)

DFS and (b) FE results. ................................................................................................... 21

Figure 2.4. Leakage flux distribution along the height of the core window in the (a)

radial and (b) axial directions.......................................................................................... 22

Figure 2.5. The 3D model with axi-symmetrical ferromagnetic boundaries. ................. 23

Figure 2.6. Comparison between the 2D and 3D axially symmetric models of leakage

flux density in the LV winding in the (a) radial and (b) axial directions. ....................... 24

Figure 2.7. Comparison between the 2D and 3D axially symmetric models of leakage

flux density in the HV winding in the (a) radial and (b) axial directions. ...................... 24

Figure 2.8. Transformer models used in the calculation of the EM forces: (a) a 3D

model with asymmetric boundary conditions and (b) a 3D models within a metal tank.

......................................................................................................................................... 26

Figure 2.9. Comparison of EM forces in LV winding in the (a) radial and (b) axial

directions. ........................................................................................................................ 27

Figure 2.10. Comparison of EM forces in HV winding in the (a) radial and (b) axial

directions. ........................................................................................................................ 27

Figure 2.11. Leakage flux distribution of a 2D axi-symmetric ¼ model. ....................... 28

Figure 2.12. Vector analysis of the leakage flux distribution in the 2D model (solid line),

3D model (dot-dashed line), and 3D model within a tank (dashed line). ....................... 29

vii

Figure 2.13. Shunts adopted in the simulation. ............................................................... 31

Figure 2.14. Influence of strip shunts on the EM forces acting on the LV winding in the

(a) radial and (b) axial directions. ................................................................................... 32

Figure 2.15. Influence of strip shunts on the EM forces acting on the HV winding in the


Figure 2.16. Influence of lobe shunts on the EM forces acting on the LV winding in the


Figure 2.17. Influence of lobe shunts on the EM forces acting on the LV winding in the


Figure 3.1. Vibration sources of a typical power transformer. ....................................... 40

Figure 3.2. CAD model of the 10-kVA power transformer. ........................................... 43

Figure 3.3. Test specimen laminated by SiFe sheets. ..................................................... 45

Figure 3.4. Input mobility of the test specimen in the in-plane direction. ...................... 46

Figure 3.5. Input mobility of the test specimen in the out-of-plane direction. ............... 47

Figure 3.6. Schematics of winding structure homogenisation used in the FE analysis. . 48

Figure 3.7. The simplified transformer winding model in one disk................................ 49

Figure 3.8. FE model of the 10-kVA single-phase transformer. ..................................... 50

Figure 3.9. Images of the test rig used in the measurement. ........................................... 51

Figure 3.10. Locations of the point force (D1 and D3 in the +Y direction, D2 in the +X

direction, D4 in the +Z direction) and vibration measurement locations. ...................... 52

Figure 3.11. Reciprocity test between driving and receiving locations: (a) D1 and T01

and (b) D1 and T07. ........................................................................................................ 53

Figure 3.12. Spatially averaged FRF of the distribution transformer. ............................ 54

Figure 3.13. Radial FRFs at the (a) T40 and (b) T45 measurement positions. ............... 55

Figure 3.14. FRFs of the power transformer around 450 Hz and its envelope. .............. 57

viii

Figure 3.15. Comparison of the core-controlled modes in the out-of-plane direction

between the test and calculated results at (a) 35 Hz, (b) 77 Hz, (c) 103 Hz, and (d) 192

Hz. ................................................................................................................................... 60

Figure 3.16. Comparison of the core-controlled modes in the in-plane direction between

the test and calculated results at 1114 Hz. ...................................................................... 61

Figure 3.17. Comparison of the winding-controlled modes at (a) 229 Hz, (b) 420 Hz, (c)

533 Hz, and (d) 683 Hz in both radial and axial directions. ........................................... 62

Figure 3.18. Comparison of the core-winding coupled modes at (a) 11 Hz, (b) 44 Hz, (c)

57 Hz, and (d) 154 Hz. .................................................................................................... 64

Figure 3.19. Comparison of the FRFs between FE and impact test results. ................... 66

Figure 3.20. Schematics of types of winding damage introduced to the FE model........ 67

Figure 3. 21. Comparison of the modal shapes of normal and damaged windings (dot-

dashed line marks the centre of the winding). ................................................................ 69

Figure 4.1. The actual experimental setup for obtaining the electrically excited FRFs. 77

Figure 4.2. Spatially averaged FRF of the distribution transformer subject to a

mechanical excitation. ..................................................................................................... 78

Figure 4.3. Bode diagrams of the mechanically excited FRF at test point T01. ............. 79

Figure 4.4. Bode diagrams of the mechanically excited FRF at test point T40. ............. 79

Figure 4.5. Bode diagrams of the mechanically excited FRF at test points T25 and T33.

......................................................................................................................................... 80

Figure 4.6. Mode shapes at the corresponding resonance frequencies. .......................... 81

Figure 4.7. Predicted natural frequencies and mode shapes of the model transformer. . 82

Figure 4.8. The spatially averaged FRF of the transformer vibration due to electrical

excitation. ........................................................................................................................ 85

ix

Figure 4.9. Spatially averaged FRFs of the transformer vibration due to electrical

excitation at different RMS flux densities. ..................................................................... 87

Figure 4.10. Magnetising curves of the model transformer at five different frequencies.

......................................................................................................................................... 87

Figure 4.11. Spatially averaged FRFs of the transformer vibration with clamping

looseness under (a) mechanical and (b) electrical excitations. ....................................... 90

Figure 4.12. The mechanically excited FRFs at (a) the core and (b) the winding of a 110

kV/50 MVA power transformer. ..................................................................................... 91

Figure 5.1. The design of longitudinal insulation and the arrangement of missing

insulation spacers as a cause of mechanical faults. ......................................................... 99

Figure 5.2. Spatially averaged FRFs of the transformer vibration due to (a) mechanical

and (b) electrical excitations with core clamping looseness. ........................................ 100


and (b) electrical excitations with winding clamping looseness. .................................. 104


and (b) electrical excitations with missing insulation spacers. ..................................... 107

Figure 5.5. Spatially averaged FRFs of the transformer vibration due to winding

looseness in the (a) radial and (b) axial directions. ....................................................... 110

Figure 5.6. Spatially averaged FRFs of the transformer vibration due to missing

insulation spacers in the (a) radial and (b) axial directions. .......................................... 113

Figure 6.1. Schematics of the OMA-based transformer condition monitoring technique.

....................................................................................................................................... 120

Figure 6.2. Time-frequency spectra of a 10-kVA transformer in (a) energising, (b)

steady, and (c) de-energising states. .............................................................................. 126

x

Figure 6.3. Vibration waveform of a 10-kVA transformer in the (a) energising, (b)

steady-state, and (c) de-energising conditions. ............................................................. 127

Figure 6.4. Time-frequency spectra of a 15-MVA transformer in the (a) energising, (b)

steady, and (c) de-energising states. .............................................................................. 128

Figure 6.5. Vibration waveform of a 15-MVA transformer in the (a) energising, (b)

steady-state, and (c) de-energising conditions. ............................................................. 129

Figure 6.6. Identified natural frequencies in the stabilisation diagram of a 10-kVA

transformer. The solid line is the spatially averaged PSD. ........................................... 130

Figure 6.7. The steady-state vibration, filtered response, and calculated correlation

function of a 10-kVA transformer. ............................................................................... 131


transformer. The solid line is the spatially averaged PSD. ........................................... 132

Figure 6.9. Identified natural frequencies in the stabilisation diagrams of a 10-kVA

transformer with core looseness. The solid line is the spatially averaged PSD without

clamping looseness. ...................................................................................................... 134


transformer with winding looseness. The solid line is the spatially averaged PSD

without clamping looseness. ......................................................................................... 135

xi

List of Tables

Table 2.1. Technical specifications of the 10-kVA small-distribution transformer. ...... 20

Table 3.1. Material properties of the transformer parts. ................................................. 50

Table 3.2. Classification of the first eighteen modes of the small-distribution

transformer ordered by classification type. ..................................................................... 57

Table 3.3. Natural frequency shifts of the winding-controlled modes due to winding

deformations (Hz). .......................................................................................................... 68

Table 4.1. Comparison of the natural frequencies of the model transformer under

supported-clamped and supported-free boundary conditions. ........................................ 84

Table 4.2. Level differences of the 2nd, 3rd, and 4th peak responses with respect to the 1st

peak response. ................................................................................................................. 85

Table 5.1. Quantitative variation of the transformer vibration FRFs due to core

clamping looseness........................................................................................................ 101

Table 5.2. Quantitative variation of the vibration FRFs due to winding clamping

looseness. ...................................................................................................................... 105

Table 5.3. Quantitative variation of the transformer vibration FRFs due to missing

insulation spacers. ......................................................................................................... 108

Table 5.4. Natural frequency shifts ( nf ) of the winding-controlled modes due to

looseness of the winding clamping force. ..................................................................... 111

Table 5.5. Natural frequency shifts ( nf ) of the winding-controlled modes due to

missing insulation spacers. ............................................................................................ 112

xii

Table 6.1. Comparison of natural frequencies (in hertz) from EMA and free-vibration-

based OMA. .................................................................................................................. 131

Table 6.2. Comparison of the natural frequencies (in hertz) from EMA and forced-

vibration-based OMA. .................................................................................................. 133

xiii

Acknowledgements

“It is a long journey with ensuing obstacles to pursue a Doctor’s degree.” I never really

understood these words until I was fully involved. Fortunately, I had a knowledgeable

and enthusiastic supervisor, who always stood behind me to deliver guidance,

inspiration, and personal help. I would like to express my deepest gratitude to him,

W/Prof. Jie Pan, for his continuous encouragement, support, and care during my PhD

study and daily life. The times we spent in the transformer laboratory, in the anechoic

chamber, and at Delta Electricity, Western Power, Busselton Water, and the Water

Corporation are sincerely cherished. I would also like to thank him for sharing the well-

equipped vibro-acoustic laboratories and providing precious field-test opportunities.

They are not only valuable to the completion of this thesis, but also beneficial to my

future career.

Special thanks go to Mr Ming Jin, who is my cater-cousin in this long journey.

Our times spent together debugging the LabVIEW programmes, preparing industry

demonstrations, and conducting field tests are memorable. The fellow group members

in the transformer project, Ms Jing Zheng, Dr Hongjie Pu, and Dr Jie Guo, are also

sincerely acknowledged for their technical support and stimulating discussions.

I also want to thank Ms Hongmei Sun and all other lab mates, visiting scholars,

exchange PhD students, and friends in Western Australia for their immeasurable help.

Thanks also go to Dr Andrew Guzzommi for proof reading this thesis.

The financial support from the China Scholarship Council, the University of

Western Australia, and the Cooperative Research Centre for Infrastructure and

Engineering Asset Management is gratefully acknowledged.

Finally, I want to thank my wife, parents, and brothers for their understanding

and encouragement throughout the entire phase of this thesis.

xiv

Declaration of Authorship

I, Yuxing Wang, declare that this thesis, titled “TRANSFORMER VIBRATION AND

ITS APPLICATION TO CONDITION MONITORING”, and the work presented herein

are my own. I confirm that:

This work was done wholly or mainly while in candidature for a research degree at this

University.

Where any part of this thesis has previously been submitted for a degree or any other

qualification at this University or any other institution, this has been clearly stated.

Where I have consulted the published work of others, this is always clearly attributed.

Where I have quoted from the work of others, the source is always given. With the

exception of such quotations, this thesis is entirely my own work.

I have acknowledged all main sources of help.

Where the thesis is based on work done jointly by myself and others, I have made clear

exactly what was done by others and what I have contributed myself.

Signed:

Yuxing Wang W/Prof. Jie Pan

(Candidate) (Supervisor)

1

Chapter 1 General Introduction

1.1 Introduction

Power transformers can be found throughout modern interconnected power systems. In

a large power grid, there can be hundreds of units in the sub-transmission and

transmission network (>120 kV) ranging from a few kilovolt-amperes to several

hundred megavolt-amperes. For an electricity company, abrupt malfunctions or

catastrophic failure of these transformers may result in direct loss of revenue. Apart

from the repair or replacement costs, indirect losses for electricity customers, i.e., the

manufacturing industry, can be very large. The potential hazards of transformer failure

are another concern, namely explosions and fires that would cause environmental

pollution.

In this context, research efforts focussing on monitoring transformer health

status have been made to prevent catastrophic incidents and prolong a transformer’s

service life. Indeed, there are various approaches based on chemical, electrical, and

mechanical mechanisms to estimate a transformer’s health status, i.e., oil quality testing,

electrical parameter measurement, thermography, and vibration-based methods. Since

on-site transformers are typically operated at high voltage, direct access to the power

distribution system and the transformer internal parts is not permitted owing to the

inherent risks. Compared with the other methods, such as Frequency Response Analysis,

Dissolved Gas Analysis, and Return Voltage Method, the vibration-based method is

more convenient and suitable for online implementation owing to its non-intrusive

2

nature. In this thesis, the investigations mainly focus on the use of vibration-based

methods.

Similar to the other monitoring approaches, a clear understanding of transformer

behaviour is important to the development of vibration-based monitoring strategies.

Specifically, a clear comprehension of the transformer vibration mechanisms and the

causes of failure would be beneficial to vibration-based monitoring methods. To

localise the damage, special attention should be paid to the vibration features induced by

structural failure. With respect to the transformer vibration system, the following three

main aspects will be examined in this thesis. They are: 1) the excitation source, which is

composed of electromagnetic (EM) and magnetostrictive (MS) forces in the active parts;

2) the frequency response function (FRF), which is determined by the transformer

structure and its supporting boundaries; and 3) the resulting vibration response, which is

typically the direct technical parameter employed for transformer condition monitoring.

First of all, the modelling of EM force in a transformer is discussed with the aim

of improving its calculation accuracy. It is well known that the interaction between a

transformer leakage field and its load currents generates EM forces in the winding.

Under this force, the power transformer vibrates and the winding experiences a high

stress burden during a short circuit. An accurate evaluation of the leakage magnetic field

and the resultant forces on power transformer windings are certainly of crucial

importance both to transformer vibration modelling and to winding strength calculation

during transformer design. However, the amplitude distribution of the EM forces varies

greatly with different winding topologies owing to the diversity of transformer designs

and custom manufacturing. Power transformers are typically composed of windings,

magnetic circuits, insulation, and cooling systems, as well as compulsory accessories

including bushings and tap changers. The arrangement of windings and ferromagnetic

3

parts will affect the leakage flux distribution as well as the EM forces. To improve the

accuracy of modelling EM force, understanding these influential factors and their

effects on the force distribution is required.

In addition to the modelling of forces in a transformer, the dynamic behaviour of

the transformer structure is investigated. Experience from transformer field tests

suggests that the vibration response of an in-service transformer may vary dramatically,

even for transformers with the same technical specifications. Inverse methods, i.e.

system identification techniques, have been used to extract the system parameters of in-

service transformers. Transformer models were considered with current, voltage, and

temperature inputs in these methods [2–4], where unknown parameters in the models

were finally determined by fitting the measured data. However, they are not capable of

providing the detailed mechanisms involved in transformer vibration. The main reason

is their inability to describe a complex system with limited variants employed in the

transformer model. Unlike vibration modelling based on the inverse method, analytical

modelling of transformer vibrations faces overwhelming obstacles due to the structural

complexity. Current understanding of this complex vibration system has not yet met

industry requirements or at least is not able to adequately guide vibration-based

monitoring methods. There is thus a need for transformer vibration modelling and

simulation approaches to provide deeper understanding of the vibration mechanisms.

In this thesis, a 10-kVA single-phase transformer is modelled based on the FE

method. When the FE method is applied, certain simplifications of the transformer core

and winding are made and justified. The vibration features of the transformer with

winding deformations are studied numerically based on a verified FE model.

The inverse method involves generating runs starting from the initial state, and removing states incompatible with the reference values by appropriately refining the current constraint on the parameters. The generation procedure is then restarted until a new incompatible state is produced, and so on iteratively until no incompatible state is generated [1].

4

To understand the dynamic properties of a vibration system, experimental

investigation is a powerful tool, especially when certain parts of the system are difficult

to deal with numerically. A detailed discussion of the modal parameters of a transformer

based on an impact test is given in this work.

Transformers are self-excited by the EM and MS forces, which are distributed

forces generated by the electrical inputs. The vibration signal employed in transformer

condition monitoring is a frequency response generated by the electrical inputs. As a

result, the FRF for this case is called the electrically excited FRF. Unlike the

mechanically excited FRF, the electrically excited FRF includes the contribution from

both the mechanically and electrically excited FRFs. In comparison with the intensive

discussions on the mechanically excited FRF, there has been a lack of study on the

electrically excited FRF associated with transformer vibration. Therefore, this thesis

also investigates the properties of the electrically excited FRF.

Assuming that the vibration features of a healthy transformer are identified, any

deviations from those features may be used as an indicator of changes in the transformer

health status and even of potential structural damage. A more challenging task is to

ascertain the type and position of the damage. To achieve this goal, a study on changes

in vibration caused by different types of structural damage is necessary. Once the

vibration characteristics of certain common types of damage are obtained and saved in a

database, a diagnostic tool can be developed to detect the types and locations of the

damage. In this thesis, correlation and causation between the changes in vibration and

types of structural damage are investigated. In particular, looseness in the transformer

winding and core and damage to the insulation are investigated experimentally in a 10-

kVA distribution transformer.

5

With the summarised vibration features and their correlations to various types of

structural damage in a transformer, the final step of the technique would be to

successfully extract them from the output vibration data. This is the step of modal

parameter identification based on the transformer’s vibration response. In the final part

of this thesis, a time-domain method is employed to identify the transformer’s modal

parameters by using data from the features of the transformer’s operating events (e.g.,

de-energisation state).

1.2 Thesis Focus

This thesis focusses on investigating transformer vibration and developing vibration-

based transformer monitoring strategies. The scope of this research covers experimental

and numerical studies on the excitation of transformer vibration, the dynamic

characteristics of the structural, and their variation in the presence of different types of

structural damage. Particular attention is paid to the EM force in the transformer

winding, since it not only excites transformer vibration, but also causes winding damage,

i.e., local deformation. Another research goal is to achieve a comprehensive

understanding of transformer vibration based on experimental modal analysis and

vibration modelling. Experimental analysis and numerical modelling of a damaged

transformer are also within the scope of this thesis. A final and important part of the

thesis is the extraction of the transformer’s vibration features from the measured

response data.

The primary goal of this research is to investigate transformer vibration to better

facilitate online condition monitoring. For practical application, successful feature

extraction from the response data is of paramount importance. Therefore, the overall

objectives of this thesis are:

6

1. To evaluate existing approaches and available literature on transformer vibration

analysis and condition monitoring.

2. To obtain analytical and numerical models for calculation of EM force in a

transformer in both 2D and 3D scenarios with and without asymmetric

ferromagnetic boundaries.

3. To study the effects of magnetic shunts and other ferromagnetic arrangements on

EM forces in transformer winding.

4. To develop numerical models for vibration analysis of core-form power

transformers, which could involve modelling complex structures such as core

laminations and winding assemblies.

5. To investigate the characteristics of transformer vibration by means of vibration

modal tests, which extend transformer modal analysis to a new level.

6. To explore the FRFs of transformer vibration experimentally, in particular, the

electrically excited FRFs, which are directly related to the vibration response.

7. To study the changes in vibration induced by structural damage in a transformer,

based on the numerical and experimental methods.

8. To analyse the features of transformer vibration and adopt them for extraction

vibration behaviours, which can be directly applied to transformer condition

monitoring.

1.3 Thesis Organisation

This thesis is orgnised as follows:

Chapter 1 serves as a general introduction, which states the research problems as

well as the specific aims and overall objectives of the thesis. Although the general

introduction illustrates the research motivations for the entire thesis, the literature and

7

research activities are reviewed in each chapter, making them self-contained and

directly relevant to those chapters.

Chapter 2 covers objectives #2 and #3 and mainly focusses on the modelling of

EM forces in a transformer. In this part, the FE method is adopted as the primary

approach to examine the influential factors that affect the calculation of force in a

complex ferromagnetic environment. An analytical method, namely the Double Fourier

Series (DFS) method, is employed to verify the FE model. Different modelling

simplifications and ferromagnetic boundaries are analysed in the finite element

calculations.

Modelling of transformer vibration is studied in Chapter 3, where the FE method

is employed in the numerical modelling. To reduce the D.O.F. of the transformer model,

appropriate simplifications are adopted, which are verified through specially designed

material tests. An experimental modal analysis is used to verify the vibration model and

to discuss its modal parameters. Vibration characteristics of a transformer with winding

faults were also investigated based on the verified FE model. Objective #4 is achieved

in this chapter.

Experimental study on the vibration response of a test transformer is introduced

in Chapter 4, in which objectives #5 and #6 are covered. From the comparison of

mechanically and electrically excited FRFs, the concept of electrical FRF in the

transformer structure is explored. Case studies related to different boundaries and a 110-

kV/50-MVA 3-phase power transformer are conducted to verify the experimental

observations.

Structural faults, i.e., winding looseness, are introduced into the test transformer.

The variations of the FRFs of transformer vibration are analysed in detail and

8

corroborated with different structural faults and severities of damage in Chapter 5,

where objective #7 is included.

Chapter 6 covers objective #8 and examines the features of transformer vibration

and transient vibrations triggered by operational events. A time-domain OMA-based

algorithm (NExt/ITD) is employed to identify the modal parameters of a 10-kVA

transformer. The features of transformer vibration and operating conditions are

considered in the proposed algorithm, which improves identification accuracy in some

cases. Identification is also achieved with the same transformer with core and winding

anomalies.

Finally, the conclusions of this thesis are given in Chapter 7, which also

provides a brief outlook for future works.

9

Chapter 2 Accurate Modelling of Transformer Forces

2.1 Introduction

With an expanding state power network, transformers with higher voltage ratings and

larger capacity are commonly utilised to satisfy the growing demands and long-distance

transmission. Therefore, load currents carried in the electrical circuit increase inevitably.

As a consequence, the EM forces generated by the interaction of the transformer

leakage field and load currents are increased. Under this force, the power transformer

vibrates and experiences harmonic loads. Since the EM forces are proportional to the

square of the load current, forces generated during a short circuit or energisation

operation may be as high as thousands to millions of newtons. In these cases, the

transformer’s vibration response increases dramatically, as does the winding stress

burden. As the resulting EM force becomes larger, the absolute error introduced by the

modelling procedure, i.e., oversimplification of the practical model, will become more

pronounced. In this context, an accurate evaluation of the leakage magnetic field and the

resulting forces on power transformer windings are important to the calculation of

transformer vibration and winding strength during transformer design.

Due to the diversity of transformer design and custom manufacturing, the

amplitude distribution of the EM forces varies greatly for different transformer

topologies. Nevertheless, transformers are typically composed of windings, magnetic

circuits, insulation, and cooling systems, as well as compulsory accessories including

bushings and tap changers. The arrangement of windings and ferromagnetic parts will

10

affect the leakage flux distribution as well as the EM forces. Understanding these

influential factors and their effects on EM force distribution is useful for accurate

prediction of the EM forces. This chapter discusses the above topics with the aim of

improving the accuracy of EM force modelling.

2.2 Literature review

The accurate calculation of EM forces is a prerequisite to accurate modelling of

transformer vibration and dynamic strength. How to accurately calculate the EM forces

is therefore a topic of vital importance to transformer designers. The study of leakage

flux and its resulting EM force has been a topic of intense research since the invention

of the power transformer. Early methods for EM force modelling were based on

simplified assumptions that the leakage field is unidirectional and without curvature.

These methods would inevitably lead to inaccurate estimates of the EM force, especially

in the axial direction at the winding ends.

The Double Fourier Series (DFS) method, which was first proposed by Roth in

1928, improved the calculation by transforming the axial and radial ampere-turn

distributions into a double Fourier series [5]. Accordingly, in 1936, Roth analytically

solved the leakage flux field for the two-dimensional axi-symmetric case by considering

proper boundary conditions [6]. Over the following decades, the DFS method was

utilised to calculate the leakage reactance, short-circuit force, and so forth [7, 8].

In order to obtain detailed information about the leakage flux distribution,

especially at the winding ends, considerable attention has been paid in recent years to

the finite element (FE) and finite difference methods [9–11]. Silvester and Chari

reported a new technique to solve saturable magnetic field problems. This technique

permitted great freedom in prescribing the boundary shapes based on the FE method [9].

11

Andersen was the first to develop an FE program for the axi-symmetric field in a 2D

situation [10, 11]. In his research, the leakage flux density of a 2D transformer was

calculated under harmonic excitations. The calculated results were used to estimate the

reactance, EM forces, and stray losses. In that same year (1973), Silvester and Konrad

provided a detailed field calculation based on a 2D technique with higher order finite

elements [12]. They concluded that the use of a few high-order elements, with direct

solution of the resulting small-matrix equations, was preferable to an iterative solution

of large systems of equations formed by first-order elements. The significant advantages

of the FE method in prescribing the boundary shapes were verified by their case studies.

Guancial and Dasgupta [13] pioneered the development of a 3D FE program to

calculate the magnetic vector potential (MVP) field generated by current sources. Their

program was based on the extended Ritz method, which employed discrete values of the

MVP as the unknown parameters. Demerdash et al. [14, 15] also contributed to the

development of the 3D FE method for the formulation and solution of 3D magnetic

field problems. In their studies, the MVP in 3D was involved in the static field

governing equation. Experimental verification of the FE results was conducted in their

later work [15] and excellent agreement with the calculated flux density was found.

Mohammed et al. [16] further demonstrated that the 3D FE method was capable of

dealing with more complex structures, i.e., the example transformers and air-cored

reactors used in their study. Kladas et al. [17] extended this method to calculate the

short-circuit EM forces of a three-phase shell-type transformer. The numerical results

were verified by means of leakage flux measurements. As a result of all this work, it

became popular to use the FE method to investigate the magnetic leakage field and

resulting EM forces of current-carrying conductors.

12

Modelling techniques and influential factors in the calculation of leakage field

were discussed in Refs. [18–22]. Amongst these works, research focussed on improving

computation time and accuracy. Salon et al. [23] discussed a few assumptions in the

calculation of transformer EM forces based on the FE method for a 50-kVA shell-type

transformer. They claimed that the introduction of a nonlinear magnetisation curve (BH

curve) in the iron did not have any significant impact on the forces acting on the coils.

The changes in the current distribution, induced by the conductor skin effect, had a

direct influence on the EM force distribution but no influence on the total force. Coil

displacement and tap changer operation may result in major changes in the flux pattern

and unbalanced forces [23]. In 2008, Faiz et al. [24] compared the EM forces calculated

from 2D and 3D FE models, and found considerable differences between them.

However, no further explanations on these differences were provided.

Briefly summarize the above literature analysis, there is an obvious trend in

using the FE method for transformer winding EM force calculation. Although the

influence factors of this force, i.e., modelling assumptions, winding geometry and

configurations have been discussed, the physical reasons causing these differences are

still unclear and not explained sufficiently. In addition, due to the difficulty in

measurement of distributed EM forces, verification for the FE calculation is another

aspect, which has not been thoroughly addressed.

In this chapter, Section 2.4.1 is dedicated to verifying the shortcomings of the

2D FE method in a 10-kVA transformer. The underlying reason for these computational

differences is discussed using magnetic field analysis. The EM forces in the transformer

winding are modelled using the DFS and FE methods, to ascertain the confidence of

each modelling method.

13

In addition to the aforementioned topics, including modelling assumptions and

calculation methods, understanding the factors influencing the EM force is also useful

for improving modelling accuracy. Indeed, previous works [25–30] have discussed

factors influencing the calculation of EM force, i.e., winding deformation, axial

movement, ampere-turn unbalance, and tap winding configurations. These were all

confirmed to affect the determination of EM forces in the winding. Cabanas et al. [28,

29] suggested using these observations in transformer condition monitoring to detect

winding failures through leakage flux analysis. Andersen [11] roughly studied the effect

of shunts on the magnetic field in terms of flux line distribution. The focus of his work

was to investigate the reduction in transformer stray loss by introducing an aluminium

strip shield into a 2D FE model. In 2010, Arand et al. [31] reported that the position,

magnetic permeability, and geometric parameters of the magnetic flux shunt had

significant effects on the leakage reactance of the transformer. A parametric study of the

effect of shield height on EM forces in a transformer was reported [22], where a 1.6-m-

high strip shunt was found to have the best shielding effect for an 8000-kVA/35-kV

power transformer.

Currently, magnetic shunts are generally adopted to reduce the leakage reactance

and power losses and to avoid overheating of metal parts in large power transformers.

Materials with high conductivity or magnetic permeability are widely used in magnetic

shunts [33]. Since a highly conductive shield will inevitably generate heat within the

transformer enclosure and then induce extra further rise in temperature, it is not

commonly adopted in practice. L-shaped, strip, and lobe shunts are three types of

magnetic shunt employed in practical transformers [34]. The L-shaped shunt is used to

enclose the corners and edges of a transformer core while the other two types are

exclusively designed for transformer windings. More details about these magnetic

14

shunts can be found in Section 2.4.2. Since the focus of this chapter is on the winding

EM calculation, only shunts near to the transformer winding are studied. The influence

of strip and lobe magnetic shunts on EM forces in a transformer will be explored using

parametric analysis.

2.3 EM force calculation using DFS and FE methods

Given the flexibility of the FE method in dealing with complex ferromagnetic

boundaries, the following discussion on the accurate calculation of EM forces is mostly

based on this method. In order to check the suitability of the FE method in modelling

EM forces of a transformer, an analytical method, namely the DFS method, is adopted

to verify the results from the FE calculation. The DFS method will be reviewed briefly

in Section 2.3.1. Since the methodology of the FE method is widely available in

textbooks on computational electromagnetics, i.e., Ref [35], calculation of the leakage

field and EM force based on the FE method will not be introduced here.

2.3.1 General formulation of the DFS method

The DFS method ingeniously takes advantage of the periodic characteristics of the

double Fourier series to deal with the ferromagnetic boundaries in transformers [5–8].

By using the MVP, the magnetic flux density can be related to the current density in

terms of a vector Poisson equation. The following deduction is a detailed introduction to

the DFS method.

According to Ampere's law [36], in a magnetostatic field, a path integration of

the magnetic field strength ( H ) along any closed curve C around an area S is exactly

equal to the current through the area, like so:

C SHdl JdS , (2.1)

where J is the current density. The right-hand term is the total current through the area

15

bounded by the curve C . Using a vector analysis of Stokes' theorem, Eq. (2.1) can be

expressed in a differential form:

rotH J . (2.2)

Considering the law of flux continuity, the magnetic flux density B satisfies the

following expression:

0S

BdS , (2.3)

which has a differential form:

0divB . (2.4)

Although the electromagnetic properties of the ferromagnetic medium are very

complicated, B and H can generally be related using the permeability :

B H . (2.5)

Typically, the permeability of a non-ferromagnetic medium has a constant value. For

the silicon-iron (SiFe) material used in power transformers, it can be a nonlinear

function of the magnetic intensity.

From vector analysis, a field vector with zero divergence can always be

expressed as the curl of another vector. In order to satisfy Eq. (2.4), the MVP A can be

defined as:

B rotA . (2.6)

According to Helmholtz’s theorem, the divergence of the vector A should be defined to

uniquely determine vector A . In order to facilitate the solution of vector A , one usually

uses the Coulomb specification as follows:

0divA . (2.7)

Hence, a differential equation about vector A can be satisfied:

1( )rot rotA J

. (2.8)

16

For a linear medium, const , the above equation can be simplified to:

rot rotA J . (2.9)

By taking into consideration:

rot rotA grad divA A (2.10)

and combining with Eq. (2.7), the vector Poisson equation for a magnetic field can be

obtained as:

A J . (2.11)

For a Cartesian coordinate system:

2 2 2x y zA i A j A k A , (2.12)

where i, j, and k are the unit vectors in the x, y, and z directions, respectively.

The vector Poisson equation (2.11) can be decomposed into three scalar

equations:

2 2 22

2 2 2

2 2 22

2 2 2

2 2 22

2 2 2

x x xx x

y y yy y

z z zz z

A A AA Jx y zA A A

A Jx y zA A AA Jx y z

. (2.13)

Simplifying the calculation into a flat 2D situation and assuming that the direction of

current density is in the z direction indicates that the current density in both the x and y

directions is zero:

, 0, 0z x yJ kJ J J . (2.14)

Therefore, the MVP is also in the z direction ( , 0, 0z x yA kA A A ). To obtain the

MVP, it is only necessary to solve a 2D Poisson equation, like so:

2 2 22

2 2 2z z z

z zA A AA Jx y z

. (2.15)

17

The next step is to determine the current density and deal with the magnetic

boundaries. The distribution of current density is firstly relevant to the configuration of

transformer windings. For the model power transformer used in this study, the window

area, including both low-voltage (LV) and high-voltage (HV) windings, is presented in

Figure 2.1.

Figure 2.1. The 2D symmetric model of a 10-kVA small-distribution transformer.

In power transformers, the EM forces are related to the leakage field in the

winding area (see Figure 2.1). Therefore, it is only necessary to solve the leakage field

in this area for computation of the EM forces. Similar to the calculation of the magnetic

field in other cases with ferromagnetic boundaries, the effect of ferromagnetic boundary

conditions needs a special treatment. Within the window area shown in Figure 2.1, the

image method is employed to approximate the boundary effect. This method takes

advantage of the intra-regional current reflecting off each boundary back and forth.

Thus, a periodic current density distribution along the x- and y-axes is formed.

Therefore, the current density can be expanded as:

18

,1 1

cos cosz j k j kj k

J J m x n y

. (2.16)

Within the window area, the 2D governing equation can be written as:

2 2

2 2z z

zA A Jx y

. (2.17)

The MVP should also be a DFS, and thus it is assumed that:

' '

1 1( cos sin ) ( cos sin )z j j j j k k k k

j kA A m x A m x B n y B n y

. (2.18)

Considering the significant difference between the magnetic permeability of air and of

SiFe, the magnetic boundary of the interaction surface of the solution domain is defined

as:

0zAn

. (2.19)

The positions of these faces are shown in Figure 2.1. To be specific:

0 0

0, 0z z

x y

A Ax y

. (2.20)

Then, 'jA and '

kB in Eq. (2.18) have to be zero as well. Considering the boundary

conditions at x t and y h :

0, 0z z

x t y h

A Ax y

. (2.21)

Then:

( 1) , 1,2, ,

( 1) , 1,2, ,

j

k

m j jt

n k kh

(2.22)

By applying the boundary conditions to Eq. (2.18), the MVP can be expressed as:

,1 1

cos cosz j k j kj k

A A m x n y

. (2.23)

19

Substituting the MVP using Eq. (2.23) in the governing equation gives:

0 ,, 2 2

j kj k

j k

JA

m n

. (2.24)

In the solution domain, the current density is:

' '1 1 1 1 1

' '2 2 2 2 2

, ,

( , ) , ,0,

z

J a x a h x hJ x y J a x a h x h

others

. (2.25)

Multiplying Eq. (2.25) by cos cosj km x n y and substituting ( , )zJ x y in Eq. (2.16) yields:

22 2

,1

cos cos cos cosj k j k i j ki

J m x n y J m x n y

. (2.26)

Equation (2.26) is integrate twice to obtain ,j kJ :

2' '

1

2' '

1,2

' '

1

4 (sin sin )(sin sin ), 1, 1

2 (sin sin )( ), 1, 1

2 ( )(sin sin ), 1, 1

0, 1, 1

i k i k i j i j iik j

i k i k i i iikj k

i i i j i j iij

J n h n h m a m a j kh t n m

J n h n h a a k jhtnJ

J h h m a m a j khtm

j k

. (2.27)

Together with Eq. (2.23) and Eq. (2.24), the MVP can be solved. Finally, the magnetic

flux density can be obtained using Eq. (2.6). The magnetic flux density becomes:

,1 1

cos sinx k j k j kj k

B n A m x n y

,

,1 1

sin cosy j j k j kj k

B m A m x n y

. (2.28)

2.3.2 Comparison of the DFS and FE methods

In this section, the EM force of a 10-kVA single-phase small-distribution transformer is

calculated using the DFS and FE methods in a 2D situation in order to compare results.

The technical specifications of this small-distribution transformer can be found in Table

20

2.1. There are a total of 240 turns of HV winding and 140 turns of LV winding in the

model transformer with outer diameters of 265 mm and 173 mm, respectively. Both the

HV and LV windings are divided into 24 disks. For computation of the EM forces, the

force density ( dF ) in the current-carrying regions are calculated by:

dF J B . (2.29)

Eventually, the resulting EM force can be obtained by integrating within the winding

column. In both calculations based on the DFS and FE methods, the geometric

parameters and material permeabilities are kept the same for consistency.

Table 2.1. Technical specifications of the 10-kVA small-distribution transformer.

Specifications Primary Secondary

Voltage [V] 415 240

Nominal current [A] 20 35

Number of disks 24 24

Total turns 240 140

Outer diameter [mm] 173 265

Inner diameter [mm] 126 210

Height [mm] 265.6 265.6

Conductor size [mm] 8×2 8×3

Approx. weight of coils [kg] 25 20

The calculated magnetic field distributions in both the radial and axial directions

are compared in Figure 2.2 and Figure 2.3. The amplitude distribution of the magnetic

flux density agrees very well in both radial and axial directions. As can be seen in

Figure 2.2, the leakage flux in the axial direction is mostly distributed between the HV

and LV windings. The maximum field strength occurs at mid-height along the winding,

where large portions of the winding area are covered by the high field strength.

21

According to Eq. (2.29), the axial flux density would generate radial EM forces in both

windings. Therefore, the maximum radial EM force is anticipated in this region since

the line current of each turn is the same.

Figure 2.2. Comparison of leakage flux density (T) in the axial direction for the (a)

DFS and (b) FE results.

Figure 2.3. Comparison of leakage flux density (T) in the radial direction for the (a)

DFS and (b) FE results.

(a) (b)

(a) (b)

22

As well as the leakage flux inducing the EM forces, the magnetic field filling the

rest of the space is also determined. As can be seen in Figure 2.2, the maximum flux

density occurs at the space between HV and LV windings. This is caused by the

superposition of leakage flux generated from two current-carrying windings with

opposite flow direction. Larger flux density is also observed at the top and bottom of the

core window.

With respect to the radial leakage flux distribution, four areas with large

magnitude can be seen in Figure 2.3. For both HV and LV windings, two sources with

opposite magnetic flux density are located at each end. Since the current flows in the

same direction, the resulting axial forces at both ends are opposite. They both compress

the winding assembly in phase at twice the operating frequency.

Figure 2.4. Leakage flux distribution along the height of the core window in the (a) radial and (b) axial directions.

A detailed comparison of the magnetic flux densities along the height of the core

window calculated by the DFS method and by the FE method is presented in Figure 2.4.

Again, a good agreement is found in the radial and axial flux densities predicted by the

two methods. Since the ampere-turn arrangement is symmetrical in both windings, the

flux density displays great symmetry along the winding height. From the above

0.05 0.1 0.15 0.2 0.25 0.3

-2

-1

0

1

2

x 10-3

Height [m]

Flu

x D

ensi

ty [T

]

FEMAnalytical

0.05 0.1 0.15 0.2 0.25 0.3

2

2.5

3

3.5x 10-3

Height [m]

Flu

x D

ensi

ty [T

]

FEMAnalytical

(b)

(a)

23

comparisons between the analytical and FE results, it appears that both the calculation

methods and the executable programs are reliable.

2.3.3 Transformer EM force calculation on a 3D symmetric model

As reviewed in Section 2.2, the EM forces calculated by the 3D FE method were

demonstrated to have better capability of simulating practical conditions. Therefore, the

following discussions are all based on 3D models. Prior to the discussion of the factors

influencing on EM forces, a verification of the 3D FE calculation procedure is required.

In this study, a 3D symmetrical model is employed for this purpose, which is shown in

Figure 2.5.

Figure 2.5. The 3D model with axi-symmetrical ferromagnetic boundaries.

The axially symmetric model in Figure 2.5 is composed of the winding

assemblies and the surrounded core, to form a symmetric magnetic flux path in the

space. The outside core is built by rotating the side limbs to form axi-symmetrical

ferromagnetic boundaries, which are identical to the assumptions of the 2D model. The

Neumann boundary condition is naturally satisfied owing to the large permeability of

the transformer core. By setting the same core permeability, the EM forces of this model

transformer are calculated using the FE method. Comparisons of the EM forces

obtained from two models are presented in Figure 2.6 and Figure 2.7 for the LV and HV

Windings

Outside core

Inner core

24

windings, respectively. The EM force is calculated in terms of the volume force at each

disk. The x-coordinate corresponds to the layer number of winding disks, where the first

layer is at the bottom and the 24th layer is on the top. The layer number used in this

thesis follows the same order unless otherwise specified.

As can be seen in Figure 2.6 and Figure 2.7, the EM forces calculated from the

2D model and the 3D symmetric model agree very well. These results suggest that the

2D model of EM force may be considered to be equivalent to the 3D model when the

ferromagnetic boundary is modelled as axi-symmetric. A 3D modelling procedure for

EM forces thus appears reasonable. More complicated cases that include asymmetric

boundaries will be introduced to the 3D FE model in further studies.

Figure 2.6. Comparison between the 2D and 3D axially symmetric models of leakage flux density in the LV winding in the (a) radial and (b) axial directions.

Figure 2.7. Comparison between the 2D and 3D axially symmetric models of leakage flux density in the HV winding in the (a) radial and (b) axial directions.

0 5 10 15 20 25-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

Rad

ial E

M F

orce

[N]

Layer

3D Symmetric Model2D Model

0 5 10 15 20 25-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Axi

al E

M F

orce

[N]

Layer


0 5 10 15 20 250.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

Rad

ial E

M F

orce

[N]

Layer


0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

Axi

al E

M F

orce

[N]

Layer


(b) (a)

(b) (a)

25

The radial EM force compresses the LV winding and simultaneously elongates

the HV winding. The force amplitude along the winding height is not uniform. It is

normally larger in the middle and gradually decreases towards both winding ends. The

axial EM force is mainly caused by magnetic flux bending at the winding tips. As

shown in Figure 2.6 and Figure 2.7, the axial EM force compresses both the LV and HV

windings. It is worth emphasising that both the radial and axial EM forces are harmonic

forces at twice the operating frequency. These forces not only induce radial vibration in

the winding, but also cause buckling and deformation when they achieve critical values.

In both the radial and axial directions, the cumulative EM forces are slightly larger in

the HV winding because of its large conductor volume. In terms of the cumulative force

density, an opposite conclusion can be drawn, which is consistent with previous

literature [21, 32].

2.4 Influential factors in modelling transformer EM forces

2.4.1 Shortcomings of the 2D model in EM force calculation

Practical transformers are typically composed of windings, a core assembly, insulation,

cooling parts, and other accessories. These transformer parts are normally not axi-

symmetric. Geometrically, it is not convenient to model the boundaries of these 3D

parts using a 2D method. Insulation and cooling parts are normally not made of

magnetic materials and therefore do not have much influence on the transformer leakage

field. However, the transformer core, magnetic shunts, and metal tank are typically

made of ferromagnetic materials. They will affect the distribution of leakage flux and

therefore EM forces.

However, in a 2D model, it is difficult to account for the complex magnetic

boundaries using the FE method since it is no longer a symmetric model with axi-

26

symmetric boundaries. Inevitably, errors will be introduced to the computation of EM

forces if the space for the leakage flux is modelled as a 2D problem. Therefore, it is

important to examine how 2D modelling will affect the accuracy of EM forces in large

transformer windings. The shortcomings of the 2D approach in modelling these

asymmetrically designed ferromagnetic parts will be discussed in this section. Figure

2.8(a) presents the active parts of a transformer including core and winding, while in

Figure 2.8(b) a metal tank is included, which is used to contain the insulation media.

These two models are employed to study their effects on the transformer leakage field

and EM forces.

Figure 2.8. Transformer models used in the calculation of the EM forces: (a) a 3D model with asymmetric boundary conditions and (b) a 3D models within a metal tank.

Figure 2.9 and Figure 2.10 present the calculated results from three FE models:

1) a 2D model, 2) a 3D model with asymmetric core, and 3) a 3D model within a metal

tank. In the following analysis, only the EM forces of the LV winding are intensively

discussed. Similar conclusions can be drawn by analysing the results for the HV

winding (see Figure 2.10) by following the same steps.

As shown in Figure 2.9, the EM forces calculated from the 2D and 3D models

deviate appreciably. A difference of 9.62% at the 1st layer in the radial direction and

(b) (a)

27

93.1% at the 16th layer in the axial direction were found, respectively. Although the EM

force distribution shows the same patterns in both directions, the calculated forces in the

2D model are larger in the radial direction and smaller in the axial direction when

compared with that of the 3D model. This comparison shows that the 2D modelling may

over-estimate the force in the radial direction and under-estimate the EM force in the

axial direction. The reason comes from the 2D model’s inability to accurately describe

the non-symmetric magnetic boundary. The 3D model is more suitable for complex

structures and boundary conditions as it is capable of better capturing the practical

situation.

Figure 2.9. Comparison of EM forces in LV winding in the (a) radial and (b) axial directions.

Figure 2.10. Comparison of EM forces in HV winding in the (a) radial and (b) axial directions.

0 5 10 15 20 25-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

EM

For

ce [N

]

Layer

2D Model3D Model3D Model within a Tank

0 5 10 15 20 25-0.06

-0.04

-0.02

0

0.02

0.04

0.06

EM

For

ce [N

]

Layer


0 5 10 15 20 250.08

0.1

0.12

0.14

0.16

0.18

EM

For

ce [N

]

Layer


0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1

EM

For

ce [N

]

Layer


(b) (a)

(b)

(a)

28

Based on the same 3D model, a metal tank is studied to account for its effect on

the EM forces. As shown in Figure 2.9, the presence of this metal tank further reduces

the axial EM force, and increases the radial EM force with respect to the 3D model.

From the above simulation results, the component relation of the EM forces can

be summarised as follows:

3 2 3

2 3 3

D D D tank

D D tank D

Fz Fz FzFr Fr Fr

. (2.30)

To explain this simulation result, a detailed magnetic field analysis (leakage reluctance

analysis) is performed. In power transformers, the portion of flux that leaks outside the

primary and secondary windings is called “leakage flux”. The intensity of leakage flux

mainly depends on the ratio between the reluctance of the magnetic circuit and the

reluctance of the leakage path [19]. Figure 2.11 shows the flux lines calculated by the

2D FE model of the 10-kVA small-distribution transformer, while Figure 2.12

highlights one of the flux lines selected at the top-right corner. Due to the symmetric

distribution of transformer EM forces, only a quarter of the leakage field in the core

window is analysed. The density of the flux lines in Figure 2.11 represents the intensity

of the magnetic field.

Figure 2.11. Leakage flux distribution of a 2D axi-symmetric ¼ model.

29

According to Fleming's left-hand rule, the amplitude and direction of EM forces

in the winding are determined by the leakage field and winding currents. Since the EM

force is perpendicular to the leakage field and the current in one coil is assumed to be

the same at different turns, the axial/radial EM force is then dependent on the

radial/axial component of the leakage flux.

As illustrated in Figure 2.11, the leakage flux lines are curved at the top of the

coils bending towards the core. They indicate that the radial component of the flux

density has a dominant role at both ends (top and bottom ends owing to symmetry),

which causes the maximum axial EM force in the winding. The flux lines at the middle

height of the winding flow almost vertical with a very small radial component,

especially in the area between the LV and HV windings. Therefore, larger radial EM

forces occur in these areas, which agree well with the calculated results shown in Figure

2.10. However, considering the different ferromagnetic boundaries introduces certain

variations in the EM force distribution. The mechanism for the ferromagnetic material

configuration to influence the leakage flux is illustrated in Figure 2.12.

Figure 2.12. Vector analysis of the leakage flux distribution in the 2D model (solid line), 3D model (dot-dashed line), and 3D model within a tank (dashed line).

Br1 Br2 Br3

Bz1

Bz2

Bz3

O Radial

Axial 2D

3DT

3D

3D

3D

3DT 2D

30

In general, the amplitude of the flux density is determined by the magnetic

reactance of the leakage path. The side limbs, together with the top and bottom yokes of

the 2D model, form a closed cylindrical path around the winding. This arrangement

provides larger space for the magnetic path with low reactance. Hence, the amplitude of

flux density in the 2D model is the largest. Since the metal enclosure provides

additional magnetic flow paths, the flux density in the 3D model within the tank is the

second largest and that of the 3D model without the tank is the smallest. With respect to

the direction of flux flow, a vector angle between the tangential direction of the flux

line and positive radial direction is defined to facilitate analysis, as seen in Figure 2.12.

In order to satisfy Eq. (2.30) and the above-discussed amplitude relation, the sequence

of vector angles in the three cases should be 2 3 3D D tank D . This indicates that the

flux lines are more prone to bending towards the top yoke in the 2D axially symmetrical

model than in the other two cases. The physical explanation is that it is due to the large

area of the ferromagnetic top yoke, which is modelled as a thick circular plate.

2.4.2 EM forces in the provision of magnetic flux shunts

In this section, the effect of the arrangement of magnetic flux shunts on the EM forces

of the power transformer is explored. Magnetic flux shunts are designed to reduce stray

losses by preventing magnetic flux from entering the ferromagnetic areas in the leakage

field. However, to avoid eddy currents in the magnetic shunts themselves, thin SiFe

sheets are typically adopted to construct shunts with different shapes. The strip-type

magnetic shunt is the most common type of shunt used in a power transformer. For oil-

immersed power transformers with 180 000 kVA capacity, the lobe-shaped shunt is

often adopted at both ends of the winding assembly [20]. Regardless of the shunt type,

they all need to be reliably earthed. In order to study their effect on transformer EM

forces, both strip and lobe shunts are considered here. These are shown in Figure 2.13,

31

where the schematic position of each shunt can be found.

Figure 2.13. Shunts adopted in the simulation.

(a) Effect of Strip-Type Shunts on EM Forces

In the first case, the effect of the arrangement of strip-type shunts on the EM forces is

studied, assuming that no other shunts are involved in the model. The magnetic shunt

considered for this analysis includes ten pieces of rectangular strips 410 mm in height,

36 mm in width, and 2 mm in depth. Their relative magnetic permeabilities are all set to

3000. Two groups of strip shunts are placed symmetrically in front of and behind the

winding assembly, as shown in Figure 2.13. The distance between the magnetic shunt

and the winding centre is used to describe the shunt position. It varies from 174 mm to

182 mm in this case study. The calculated EM forces are presented in Figure 2.14 and

Figure 2.15, where the red dashed lines (no shunt) represent the EM forces generated

from the 3D model within the metal tank.

32

Figure 2.14. Influence of strip shunts on the EM forces acting on the LV winding in the (a) radial and (b) axial directions.

Figure 2.15. Influence of strip shunts on the EM forces acting on the HV winding in the (a) radial and (b) axial directions.

The following analysis focusses on the EM forces of the LV winding in order to

avoid redundancy. The effect on the EM forces in the HV winding can be determined by

following the same analysis method, and is shown in Figure 2.15. As shown in Figure

2.14, there are no obvious changes in the EM forces when a magnetic shunt is located

close to the front and back surfaces of the metal enclosure, i.e., at position 03. In this

position, the strip shunts do not have much effect on the leakage flux distribution since

they are too close to the metal enclosure, which is also a magnetic shunt with larger

surface area. In other positions, the presence of the strip magnetic shunts reduces the

0 5 10 15 20 25-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06R

adia

l EM

For

ce [N

]

Layer

Position 01 d=174mmPosition 02 d=178mmPosition 03 d=182mmNo Shunt

0 5 10 15 20 25

-0.05

-0.03

-0.010

0.01

0.03

0.05

Axi

al E

M F

orce

[N]

Layer


0 5 10 15 20 250.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

Rad

ial E

M F

orce

[N]

Layer


0 5 10 15 20 25

-0.1

-0.05

0

0.05

0.1

Axi

al E

M F

orce

[N]

Layer


(b) (a)

(b) (a)

33

EM forces in the axial direction but increases them in the radial direction. The

underlying reason is also the change in magnetic reactance. Moving the magnetic shunts

closer to the winding physically reduces the distance between the winding assembly and

the magnetic shunts. Therefore, the length of the magnetic path is shortened which

results in a reduction of the magnetic reactance. Consequently, the amplitude of the

magnetic flux density increases when the shunts are placed closer to the winding.

Theoretically, the EM forces should increase in both directions if the vector

angle remains unchanged. However, the axial EM force is calculated to decrease

gradually. A plausible explanation is that the changes in vector angle lead to a smaller

curvature as the shunts come closer. Meanwhile, the influence on both the radial and

axial EM forces becomes more sensitive, which is confirmed by the same distance

moved in the two cases (from 174 mm to 178 mm and from 178 mm to 182 mm).

(b) Effect of Lobe-Type Shunts on EM Forces

In the second case, the effect of lobe-type magnetic shunts on the EM forces was

studied individually. As illustrated in Figure 2.13, two sets of lobe magnetic shunts

were stacked at both ends of the winding assembly at a distance of 30 mm, which equals

the thickness of insulation plates. The lobe-shaped magnetic shunt is composed of two

half rings, which cover the main leakage path between two windings. The inner radius

is 63 mm, while the outer radius varies from 132.5 mm to 142.5 mm for the parametric

study. The calculated EM force results are presented in Figure 2.16 and Figure 2.17.

34

Figure 2.16. Influence of lobe shunts on the EM forces acting on the LV winding in the (a) radial and (b) axial directions.

Figure 2.17. Influence of lobe shunts on the EM forces acting on the LV winding in the (a) radial and (b) axial directions.

A lobe shunt with the same diameters as the HV windings in the “Radius 01 r =

132.5 mm” case has a trivial effect on the EM forces in the winding. However, the

influence becomes obvious as the outer radius increases. A general tendency is that a

lobe shunt near the winding assembly is able to increase the radial EM forces while

reducing the axial EM forces. Unlike what was observed in the strip shunt case, the

most affected areas lie at both ends of the winding assembly. Moreover, the variation is

more sensitive to the changing radius in the near field. There is an approximately 11.7%

greater reduction in the maximum radial forces calculated with a 5 mm increase in

0 5 10 15 20 25-0.11

-0.1

-0.09

-0.08

-0.07

-0.06R

adia

l EM

For

ce [N

]

Layer

Radius 01 r=132.5mmRadius 02 r=137.5mmRadius 03 r=142.5mmNo Shunt

0 5 10 15 20 25

-0.05

-0.03

-0.010

0.01

0.03

0.05

Axi

al E

M F

orce

[N]

Layer


0 5 10 15 20 250.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

Rad

ial E

M F

orce

[N]

Layer


0 5 10 15 20 25-0.1

-0.05

0

0.05

0.1A

xial

EM

For

ce [N

]

Layer


(b) (a)

(b) (a)

35

radius from 132.5 mm to 137.5 mm than from 137.5 mm to 142.5 mm. Once again,

these changes can be explained by vector analysis of the leakage flux density. When the

magnetic field is affected by a pair of large lobe shunts, the magnetic reactance becomes

smaller, which results in the increase in flux density. To explain the reduction in axial

forces, the vector angle ought to increase simultaneously to produce less radial

components.

2.5 Conclusion

In order to improve the accuracy of EM force modelling in a transformer, a few

influential factors were investigated. These included asymmetric magnetic core

modelling, metal enclosures, and strip and lobe shunts. During this numerical study, the

DFS and FE methods were employed to calculate the EM forces and compare their

predictions with each other. The main achievements of this chapter are summarised as

follows:

The shortcomings of the 2D model for EM force calculation were analysed. An

asymmetric transformer core and metal enclosure must be considered in the EM

force model as they introduce non-negligible effects on the leakage flux

distribution.

The effects of various magnetic shunts on EM forces are significant, and depend

on the shunt shape, position, and dimensions. The presence of magnetic shunts,

irrespective of type, increases the radial EM force on one hand and decreases

the axial EM force on the other hand. The most affected areas are the middle

and ends of the winding assembly.

By discussing the influential factors in modelling the EM forces of a transformer, this

work is also beneficial to engineers in the transformer industry who wish to calculate

36

winding strength and design magnetic shunts.

37

Chapter 3 Modelling of Transformer Vibration

3.1 Introduction

In Chapter 2, the on-load excitation in terms of EM forces in the winding was modelled

using the double Fourier series (DFS) and finite element (FE) methods. The effects of

simplified models and configurations of the ferromagnetic materials, e.g., the magnetic

shunt arrangement, on the accuracy of the resulting electromagnetic (EM) force were

discussed. It was shown that the accuracy of the predicted EM force could be improved

through accurate modelling of the boundaries of the magnetic field. The EM forces in

the winding will generate structural vibration of the transformer and noise. Meanwhile,

vibration-based methods for transformer condition monitoring rely on the observation

of the vibration features of the transformer structure, especially the changes in those

features when certain parts of the structure possess faults. It is therefore necessary to

understand the vibration characteristics of the transformer structure and utilise them for

condition monitoring proposes. In this chapter, an FE method is used to analyse the

vibration characteristics of the transformer structure.

This chapter first discusses the FE model and its verification based on

experimental modal analysis (EMA). Then, the modal characteristics of a core-form

power transformer are studied experimentally and numerically using EMA and the FE

simulation. Artificial structural faults are also introduced into the winding, enabling a

virtual simulation of transformer faults instead of expensive factory tests. The main

purpose of this simulation is to search for indicators associated with various structural

38

faults for the purposes of vibration monitoring. As will be seen, the simulation also

helps to understand some interesting vibration phenomena in transformers.

3.2 Literature review

Transformers in operation always vibrate and emit noise. Sources of transformer

vibration include magnetostrictive forces in the transformer core, electromagnetic forces

in the windings, and other mechanical and fluidal excitations from cooling fans and

pumps [37]. The study of transformer vibro-acoustic properties can be traced back to

near the end of the 19th century, when the noise emission became an environmental

consideration [38]. In 1894, Remington firstly stated the vibration behaviour of an air-

core transformer accompanied by a discussion of its electromagnetic properties [38].

Subsequently, investigating the transformer’s audible noise, its characteristics, and

reduction techniques has been a hot topic [39–42]. Apart from the impetus from

environmental concerns, the vibroacoustic characteristics of power transformers have

recently been employed as indicators of a transformer’s health condition [43–47].

However, regardless of the requirements for low-noise design or the development of

vibroacoustics-based condition monitoring, a better understanding of transformer

vibration mechanisms is always useful.

In the field of core vibration investigation, Weiser et al. [48] summarised the

relevance of magnetostriction and EM forces to the generation of audible noise in

transformer cores based on their experimental observations. The local flux distribution

around the air gaps in core joint regions was described graphically. Unlike the

homogeneous flux distribution outside the joint regions, interlaminar magnetic flux

occurred as a result of the large magnetic resistance of the air gaps. The regional

concentration of magnetic flux led to the saturation of the silicon-iron (SiFe) sheets at

39

these areas. Saturation of the SiFe sheets was found to be an important source of

vibration harmonics and excessive noise. No-load vibration was produced by the

magnetostriction of the SiFe material and the EM forces between core laminations.

The factors influencing transformer core vibration were studied to obtain further

insights [49, 50]. Moses [49] measured the core vibration under different clamping

pressures for both old-fashioned SiFe and modern materials. The test results showed a

smaller magnitude for vibration in modern materials. However, the internal stress of the

transformer core was able to generate more vibrations. Valkovic and Rezic discussed

different joint configurations of the transformer core to find their influence on

transformer power loss and vibration [50]. They found the multi-step-lap joint

performed acoustically better than the single-step-lap configuration.

With respect to the winding vibration, the main cause can be attributed to the

EM forces resulting from the transformer leakage magnetic field and current in the

winding. These EM forces are proportional to the square of the load currents. According

to previous studies [18, 51], the resulting load vibration was predominantly produced by

axial and radial vibration of the transformer windings. The literature on the calculation

of leakage field and EM forces has been reviewed in Chapter 2 and will not be repeated

here.

Structural components made of metal materials, e.g., magnetic shunts, also

vibrate after magnetisation by the leakage field. Magnetostrictive forces in

ferromagnetic parts or EM forces in metal components with high conductivity caused

by eddy currents are the underlying driving forces. The vibration induced in these parts

occurs in both no-load and load cases. In addition to the above-mentioned vibration

sources related to the transformer magnetic circuit, excitations arising from cooling

equipment should also be included for a comprehensive understanding of transformer

40

vibration. The vibration composition of a typical transformer can be summarised as

shown in Figure 3.1.

Figure 3.1. Vibration sources of a typical power transformer.

The aforementioned studies provide a basic understanding of a transformer’s

vibration mechanisms and indeed assist transformer design to some extent, i.e., the

selection of low magnetostrictive core materials and multi-steplap lamination design for

transformer low noise optimization. However, they are still inadequate for vibration-

based transformer life management, condition monitoring, and fault diagnosis since

their realisation requires more accurate information on transformer mechanical status,

which usually cannot be determined by qualitative studies. Therefore, researchers in

these areas eagerly anticipate a more detailed understanding of the vibration

characteristics. With this in mind, theoretical methods, including analytical modelling

and numerical simulation, and elaborately conceived experiments are adopted to cater to

this demand.

The modelling of transformer core vibration may have originated from Henshell

et al.’s work in 1965 [52]. They considered the transformer core as a seven-beam

Magnetostriction

EM Forces

EM Forces

Pumps

Cooling Fans

Magnetization of Structural Components

Tran

sfor

mer

Vib

ratio

n

No-load Vibration

Load Vibration

Accessory Vibration

41

system with springs connected at square-cut or mitred joint regions. A theoretical model

was established, taking longitudinal vibrations, shear deformation, and rotational inertia

into account. In their results, the out-of-plane natural frequencies were verified to be

much lower than those in-plane.

The FE method was utilised to model the transformer core vibration [53–55].

Moritz [52] calculated the in-plane modes of a core model and used it for acoustic

prediction. Kubiak and Witczak [54, 55] analysed the force vibration caused by

magnetostriction of the transformer core using the FE method. Their core models

included the orthotropic property of the transformer core. Chang et al. published a

research paper in 2011, which presented a state-of-the-art transformer core model [56].

The actual geometry of the core cross-section, as well as the geometry of the core-type

transformer, was accounted for in their models.

The dynamic properties of transformer winding were also explored, especially in

the area of strength calculation. The axial vibration of transformer winding was

modelled in Refs. [57–59]. In these studies, the winding assembly was treated as a

spring–mass–damper system. The FE modelling technique was adopted to study the

winding axial vibration in Ref. [60]. Meanwhile, discussions on the radial vibration of

the transformer winding appear to have been rare over the past decades. Kojima et al.

[61] studied the winding radial response by calculating its buckling strength. In his

study, the transformer winding comprises curved beams with laminated structures. The

short-circuit radial response of a core-type transformer winding was calculated. The

winding assembly was modelled as a lumped-parameter system in the radial direction

[62].

In addition to modelling the transformer vibration separately, with only a core or

winding model, theoretical investigations of transformer vibration as a whole structure

42

were also conducted to obtain its coupled vibration characteristics [63–65]. Owing to

the limitation of analytical solutions for such a complex structure, numerical

simulations are currently the only alternative for investigation of transformer vibration.

Rausch et al. [63] developed a calculation scheme for the computational modelling of

the load-controlled noise. Ertl and Voss [64] illustrated the role of load harmonics in the

audible noise of electrical transformers based on a 3D FE model. They focussed on the

prediction of transformer load noise. Modelling considerations and mechanical analyses

were made by Ertl and Landes [65], where the winding mode shapes were briefly

discussed. Ertl and Landes [65] classified the transformer winding modes in general as

flexural modes, longitudinal oscillation, and mixed modes (with flexural and

longitudinal components). Representative winding mode shapes were graphically

described, although they were not related to the transformer core as a whole.

In conclusion of the literatures on vibration modelling of transformer core,

winding and their assembly, no analytical tools or generally acknowledged numerical

simulation schemes have been formed in current researches. However, the FE method

behaves as a powerful and alternative tool for this challenge. As is well-known, a

desirable FE model requires proper assumptions and boundary settings, which are not

always identical and difficult to be determined. The necessity of advancing it to a more

applicable and accurate tool becomes much urgent.

Along this direction of research, a transformer vibration model considering both

core and winding assemblies is established based on a 10-kVA distribution transformer.

To better facilitate the development of condition monitoring strategies, structural

anomalies are introduced to this model so as to ascertain their influence on the

transformer vibration features.

43

3.3 Modelling setup and strategy based on FE method

The transformer used for vibration modelling is a 10-kVA distribution transformer

based on the disc-type winding structure of a large power transformer. It was

specifically designed and manufactured by Universal Transformers, and is shown in

Figure 3.2.

Figure 3.2. CAD model of the 10-kVA power transformer.

The nominal current of this transformer is 20 A in the primary winding (240

turns) and 35 A in the secondary winding (140 turns). The transformer core is stacked

by 0.27-mm-thick grain-oriented SiFe sheets. In each joint region, overlapping is

created using the single-step-lap method with two plates per step. The core stack is

fixed in place by sets of metal brackets clamped with eight bolts. Likewise, the winding

assembly is fixed to the bottom yoke by two pressboards with four bolts. The clamping

force can be changed by adjusting these bolts.

Because of the non-axial symmetry of the transformer structure, a full 3D FE

model has to be established to examine the detailed responses without loss of basic

vibration features. To allow a full 3D calculation of the transformer vibration with

44

reasonable expenditure of calculation time, the FE discretisation of the core and

winding assemblies needs to be specially designed.

To minimise the core loss and eddy currents in the magnetic path, SiFe sheets

are adopted to compose the transformer core. The thickness of each SiFe sheet is around

0.2–0.4 mm depending on the rolling technique. Thin copper strands are employed in

the design of the transformer coils to avoid the “skin effect” of a single current-carrying

conductor. A single winding turn with a large cross section is replaced by several thin

strands with the same total cross-sectional area. The group of copper strands is then

wrapped together using a continuously transposed technique to finally form the

transformer winding. The application of SiFe sheets and transposed winding improves

the electrical performance of the transformer. However, it makes modelling the

transformer vibration more complicated. Compared to the whole structure, the

dimensions of each element of these assemblies are relatively small. The resolution of

the FE mesh must handle the thin SiFe laminations and winding conductors as well as

the large scale of the transformer tank. In this situation, the D.O.F. of the FE model

would be very large. Therefore, effective simplifications of the transformer core and

winding in the vibration model are necessary to facilitate the calculation and analysis.

3.3.1 Modelling considerations for the transformer core

It is obvious that meshing the core in terms of each SiFe lamination is unacceptable for

practical simulation. The huge number of D.O.F. in the system matrix would likely

incur severe computational costs. How to deal with this difficulty is an important

question in transformer vibration modelling. In this section, an equivalent method

considering the effective Young’s modulus of the transformer core assembly is

proposed to tackle this problem. This equivalent method is based on the experimental

modal test of a test specimen comprising SiFe sheets. Figure 3.3 shows the test

45

specimen used in the experiment. The specimen is assembled from 100 layers of 0.25-

mm-thick SiFe sheets. At each end of the SiFe sheet, a bolt hole is drilled in order to

clamp the laminations once assembled.

Figure 3.3. Test specimen laminated by SiFe sheets.

The experimental modal test was conducted in order to obtain the natural

frequencies and hence the Young’s modulus according to the Euler–Bernoulli beam

theory:

2

40

n nEILA L

, (3.1)

where 𝜔𝑛 is the natural frequency, 𝛽𝑛𝐿 is a constant referring to each mode, E is the

Young’s modulus, 𝜌 is the material density, 0A is the cross-sectional area, and I is the

area moment of inertia.

The vibration is measured in the in-plane and out-of-plane directions separately

in order to obtain the anisotropic Young’s modulus of the test specimen. The out-of-

plane direction is along the Z direction shown in Figure 3.3.

X Y

Z O

46

Figure 3.4 shows the Bode diagram of the input mobility (of the flexural wave)

in the in-plane direction (the direction parallel to the core lamination). Two resonance

peaks are clearly shown at 2638 Hz and 6448 Hz. According to Eq. (3.1), the calculated

Young’s modulus of the test specimen is E = 158.6 GPa in this direction. The Young’s

modulus of the SiFe material is approximately 180 GPa, which is of a similar order to

that measured in the in-plane direction. This indicates that the lamination of the SiFe

sheet has trivial influence on its material properties in the in-plane direction and one can

regard the core lamination as a solid entity.

Figure 3.4. Input mobility of the test specimen in the in-plane direction.

For the out-of-plane direction (the direction perpendicular to the core

lamination), the input mobility is shown in Figure 3.5, where the natural frequencies are

found to be much lower than those in the in-plane direction. If the test specimen is made

of SiFe material without lamination, then the 1st out-of-plane natural frequency should

be at 1459.2 Hz according to Eq. (3.1). However, the measured fundamental natural

frequency is merely 130.5 Hz. This remarkable difference comes from the lamination of

-50

0

50

FRF

[dB

]

0 2000 4000 6000 8000 10000 12000-200

0

200

Frequency [Hz]

Pha

se [o ]

47

the SiFe assembly, since its bending stiffness in the out-of-plane direction is reduced

significantly after lamination. Rearranging Eq. (3.1), the Young’s modulus in the out-

of-plane direction can be obtained as 1.8 GPa based on the measured resonance

frequency. This indicates a strong anisotropy in the mechanical properties of the core

assembly. In this situation, the bending stiffness is not only determined by the Young’s

modulus of a single SiFe sheet, but is also related to the interaction between sheets, e.g.,

friction due to bolt forces.

Figure 3.5. Input mobility of the test specimen in the out-of-plane direction.

3.3.2 Modelling considerations for the transformer winding

In order to carry large currents with low eddy-current losses, the winding conductors of

power transformers usually consist of several copper single strands in one turn and

continuously transposed, as shown in Figure 3.6. Due to the transposing, all strands in a

specific turn experience approximately the same amount of leakage flux. This design

satisfies the requirements of the electrical considerations, but increases the complexity

of its vibration modelling. To reduce the system of equations to be solved in the FE

-20

0

20

FRF

[dB

]

0 200 400 600 800 1000-200

0

200

Frequency [Hz]

Pha

se [o ]

48

model, the winding structures must be simplified to a homogenised 3D model with the

same shape and size. The homogenisation procedure of the winding disk is illustrated in

Figure 3.6.

Figure 3.6. Schematics of winding structure homogenisation used in the FE analysis.

To simplify the winding model, it is assumed that the materials possess an

equivalent stiffness, mass, and damping ratio. Since the coil, insulation paper, and

spacer blocks alternate in axial and radial directions, the homogenisation is first applied

within the transposed conductor, where the single strands can be regarded as both

parallel- and series connected in between. For the parallel connection, the effective

stiffness is determined by linear elastic springs representing the copper conductors and

wrapped papers; see Figure 3.6. Since, in a parallel connection, the overall stiffness is

dominated by the stiffest spring, it can be approximated by the copper stiffness, which

is much higher than that of insulation papers. The same approach to homogenisation is

applied to the series connection, where the most flexible material, i.e., the insulation

papers, dominates the stiffness. Although the individual thicknesses of insulation papers

are trivial, they contribute to the overall stiffness of the winding assembly.

49

Following this simplification, a sketch of the homogenised winding disk is

shown in Figure 3.7. Simplified winding conductors are placed tightly in sequence with

contact surfaces in between. This simplified winding structure is in fact the design for a

low-capacity transformer, i.e., the model transformer employed in this study. Similar to

the transformer core, the winding disks can be regarded as laminated with a single

copper lead. Therefore, the winding assembly exhibits similar anisotropic mechanical

properties as the transformer core. It is worth noting that the model is best treated in

cylindrical coordinates to facilitate straightforward calculation. As shown in Figure 3.7,

the direction of lamination is in the radial direction.

Figure 3.7. The simplified transformer winding model in one disk.

3.3.3 FE model of the test transformer

After accounting for the considerations of the transformer core and winding and their

geometric dimensions, the FE mesh is finally established as shown in Figure 3.8. The

FE model involves core and winding assemblies, clamping bolts and brackets, and

bottom boundary conditions, and is set forced-free (the bottom is clamped and the top is

free) to simulate the laboratory situation. In addition to the simplification of the

transformer core and windings, the material properties of the non-metallic materials, i.e.,

insulation paper, are assumed to be linear. Nonlinearity induced from moisture content

and oil saturation will not be considered in the FE model. The upcoming numerical

r s t

50

analysis will be restricted to linear mechanics, thus any dynamic changes in the material

behaviour will not be considered. Table 3.1 lists the material parameters adopted in the

FE model.

Figure 3.8. FE model of the 10-kVA single-phase transformer.

Table 3.1. Material properties of the transformer parts.

Part Material Density

(kg/m3)

Young’s Modulus

(GPa)

Poisson

ratio

Brace Steel 7850 210 0.3

Bolts Steel 7850 210 0.3

Insulation block Phenolic 820 20.72 0.4

Clamping plate PBT 900 7.69 0.48

3.4 FE model verification by means of modal analysis

Validation and updating of the FE model are usually compulsory before performing

further numerical analysis. In this study, a transformer modal test was employed to

Left limb

Top yoke

HV winding

LV winding

Bolts Pressboard

Brackets

51

validate the established FE model. This correction procedure is generally an iterative

process and involves two major steps: 1) comparison of the modal parameters obtained

from both the FE model and modal test, and 2) adjustment of the FE model to achieve

more comparable results. It should be emphasised that mode shapes have to be checked

when identifying the resonance modes.

3.4.1 Modal test descriptions for a single-phase transformer

The graphical illustration of the test-rig used in the modal test is shown in Figure 3.9,

where a 10-kVA distribution transformer is supported on the ground by two rigid blocks.

For this case, a multiple-input frequency analyser (B&K, Pulse, 3560) was used for data

acquisition and to calculate the acceleration frequency response functions (FRF) at sixty

measurement locations. The transformer vibration was excited by an impact hammer

(B&K, 8206). Six accelerometers (IMI, 320A) were used in the measurement. The

impact hammer and accelerometers were calibrated before each experiment.

The impact force location and 48 vibration measurement locations are shown in

Figure 3.10. The other twelve measurement locations are located on the back and right

sides of the model transformer and thus are not shown in Figure 3.10.

Figure 3.9. Images of the test rig used in the measurement.

52

The choice of excitation location in Figure 3.10 is based on the need to excite

the most structural modes and to do so at a location where the mode shape function will

be as large as possible. Nevertheless, the magnitude of the mechanically excited FRF

will depend on the location of the excitation.

Figure 3.10. Locations of the point force (D1 and D3 in the +Y direction, D2 in the +X

direction, D4 in the +Z direction) and vibration measurement locations.

Further experimental verification of the reciprocity② between the excitation

location and two measurement positions (T01 and T07), as shown in Figure 3.11, will

shed some light on the effect of the excitation position on the measured FRFs. A very

good reciprocity between the driving and receiving locations is observed in both cases,

indicating that a significant saving in measurement time can be achieved if the FRF is

also calculated with a spatial average of the excitation locations. On the other hand, the

difference in the FRFs shown in Figures 3.11(a) and (b) illustrates the dependence of

the mechanically excited FRF on the driving location. The results also confirm that the

locations of the natural frequencies are independent of the driving location, as expected.

② Reciprocity in the FRF measurements is defined as follows: the FRF between points p and q determined by exciting at p and measuring the response at q is the same FRF found by exciting at q and measuring the response at p (Hpq = Hqp).

53

Figure 3.11. Reciprocity test between driving and receiving locations: (a) D1 and T01 and (b) D1 and T07.

3.4.2 Modal analysis of the single-phase transformer

The spatially averaged transformer vibration FRFs in all three directions are firstly

studied in order to determine the natural frequencies; see Figure 3.12. The resonance

behaviour of the transformer structure is clearly observed, where the response

magnitude at high frequency is increasing gradually. This indicates that the high-

-35

-30

-25

-20

-15

-10

-5

0

5

FRF

[dB]

HD1->T01HT01->D1

(a)

100 200 300 400 500-60

-50

-40

-30

-20

-10

0

Frequency [Hz]

FRF

[dB]

HD1->T07HT07->D1

(b)

54

frequency vibration is easier to excite with the same input force. Restrictions on noise

emission at higher frequencies are important owing to the sensitivity of human hearing.

It is thus critical for transformer manufacturers to optimise their designs to have a

reduced high-frequency response. Additionally, the measured resonance peaks in the

low-frequency range appear denser than in the high-frequency range. The optimised

transformer design should thus be capable of avoiding resonance frequencies in the low-

frequency range in the operating condition, i.e., at 50 Hz and its harmonics.

Figure 3.12. Spatially averaged FRF of the distribution transformer.

A cluster of resonance peaks occurs in the frequency response between 400 Hz

and 500 Hz. These are unlike the other resonance peaks, which have clear and smooth

curves. Special attention is paid to this region to identify whether it is a result of

winding or core modes. After a detailed examination of the vibration FRFs at all test

points, the resonances within this frequency range were only found in the FRFs

measured on the winding assembly, particularly in the radial direction. To verify

200 400 600 800 1000 1200 1400-25

-20

-15

-10

-5

0

5

10

Frequency [Hz]

FRF

[dB]

55

whether there is more than one mode or not, the FRFs measured from two different

points in the winding radial direction are presented in Figure 3.13.

Figure 3.13. Radial FRFs at the (a) T40 and (b) T45 measurement positions.

It is noted that a 180° phase change at around 420 Hz is obvious at the T40 point.

However, three peaks occur near the same frequency at the T45 point. The phase

changes corresponding to each peak are recorded with smaller phase differences. The

200 400 600 800 1000 1200-30-20-10

010

FRF

[dB

]

200 400 600 800 1000 1200-200

0

200

Frequency [Hz]

Pha

se [o ]

200 400 600 800 1000 1200-30-20-10

010

FRF

[dB

]

200 400 600 800 1000 1200-200

0

200

Frequency [Hz]

Pha

se [o ]

(a)

(b)

56

overall phase change of these three peaks is 180° in total. This observation denotes that

local resonances occur around the global natural frequency. The reason the local

resonance is excited with different frequencies is the winding’s local stiffness. The local

stiffness varies at different measurement points and thus leads to a complex distribution

at the spatially averaged FRFs. The local resonance occurs at a wide range of FRFs. If

they occasionally occur around the global resonance frequency, then these local

resonance peaks will be more pronounced. The phenomena observed in the transformer

winding mostly agree with this situation. In addition, it is noted that the two

measurement points vibrate anti-phase, as can be seen from the Bode plot. This is useful

for producing a sketch of the mode shape, which is important for the understanding of

the transformer vibration mechanisms, and even its optimised design.

Based on the aforementioned analysis, it is confirmed that there is only one

global mode in the range from 400 Hz to 500 Hz, although a few local resonance

frequencies occur in a narrow frequency band around this range. Therefore, an envelope

FRF of these peaks would be useful to represent the modal response. Based on the

envelope FRF shown in Figure 3.14, the natural frequency of this global mode can be

determined.

57

Figure 3.14. FRFs of the power transformer around 450 Hz and its envelope.

To summarise the measured FRFs, Table 3.2 lists the first eighteen measured

natural frequencies of a core-type single-phase transformer. These modes cover a

frequency range up to 1500 Hz and their mode shapes are classified into three categories

according to the modal participation of the core and/or winding. In this regard, the

transformer modes are generally described as core-controlled, winding-controlled, and

coupled-mode. The modal distribution is core-controlled if the largest vibration

response dominating the transformer vibration occurs in the transformer core

component. The definition is similar for winding-controlled modes. The model

distribution is defined as coupled-mode if the winding and core both significantly

contribute. A graphical comparison between the measured and simulated mode shapes

is conducted in order to verify the modes in the following part.

Table 3.2. Classification of the first eighteen modes of the small-distribution transformer ordered by classification type.

Group Measured Calculated Mode shape summary

58

natural

frequency

[Hz]

natural

frequency

[Hz]

Core-controlled

35 36.7 out-of-plane torsional mode

77 82.27 out-of-plane asymmetrical bending


192 195.22 out-of-plane symmetrical bending

239 210.27 out-of-plane symmetrical bending



1140 1063.6 in-plane symmetrical bending

Winding-

controlled

229 267.71 rigid-body translation in axial direction

420 415.30 cylindrical mode (2,1)

533 554.81 axial bending



969 1004.7 radial bending

Coupled-mode

11 12.07 out-of-plane in-phase bending

44 44.71 in-plane in-phase bending

53 48.3 out-of-plane anti-phase bending

154 132 in-plane anti-phase bending

59

As can be seen from Table 3.2, the winding-controlled resonances are generally

in the higher frequency range while the core-controlled and coupling modes are usually

in the lower frequency range.

In principle, the mode shape analysis would be beneficial for a quantitative

understanding of transformer structure resonance. In the following section,

representative mode shapes of each category are analysed.

(a) Core-controlled mode

Typical core-form power transformers are constructed with SiFe sheets in a shape that

can be regarded as a joint structure composed of several “beams”. The number of beams

depends on whether it contains the side limbs acting as an additional magnetic path, i.e.,

the three-phase five-limb core transformer. The discussion here is based on a

transformer with a three-limb core and aims to reveal the common features of

transformer modes.

Figure 3.15 presents the first four core-controlled modes at 35 Hz, 77 Hz, 103

Hz, and 192 Hz. With respect to the experimental mode shapes, the vibration responses

measured at the core test points are employed, which correspond to the discrete points

in Figure 3.15. To better demonstrate the measured mode shapes, the undeformed frame

including the core yokes and limbs is overlapped in the same figures. The four beams

represent the core frame and the vertical lines in the centre represent the front and back

boundaries of the winding. In the core-controlled modes, there is no obvious winding

vibration. The two vertical lines in the middle are merely for reference.

60

Figure 3.15. Comparison of the core-controlled modes in the out-of-plane direction between the test and calculated results at (a) 35 Hz, (b) 77 Hz, (c) 103 Hz, and (d) 192

Hz.

Generally speaking, the FE calculated mode shapes agree well with those

measured in the modal test. The maximum frequency deviation is 13.6% for the 5th

core-controlled mode. As can be seen in Figure 3.15, the first four modes are out-of-

plane core modes, which are in a low-frequency range. Judging from their mode shapes,

they are obviously not the rigid-body modes. As mentioned in Section 3.3.1, the out-of-

plane modulus is much lower than the in-plane modulus. This would be the underlying

reason for these low-frequency resonances. For the 1st mode, the transformer core is

twisting symmetrically. The 2nd and 3rd modes are related to the asymmetric bending of

the right and left limbs, respectively. Starting from the 4th mode, the core-controlled

modes are all related with complex bending in the out-of-plane direction. It can be

expected that the five-limb core-form transformer will also include such out-of-plane

modes. Although they will be at different frequencies, they would be in the low-

frequency range as well.

(a) (b)

(c) (d)

61

The natural frequencies of the in-plane modes are expected to be higher. There is

only one in-plane mode measured below 1500 Hz, which is shown in Figure 3.16. This

mode is related to the in-plane bending of the side limbs. The above analysis verifies

that the core assembly is no longer a simple steel frame with isotropic material

properties. The anisotropic properties of the transformer core allow many more out-of-

plane resonances in the low-frequency range. This feature is important in transformer

noise abatement, which can be achieved by optimising the structure design to alter the

natural frequencies in the low-frequency range.

Figure 3.16. Comparison of the core-controlled modes in the in-plane direction between the test and calculated results at 1114 Hz.

(b) Winding-controlled mode

With respect to the winding-controlled modes, it is worth emphasising that practical

power transformer windings are mostly wound in a cylindrical shape. The following

discussion on winding-controlled modes may be of general significance under this

consideration. Similar to the definition of the core-controlled mode, the winding-

controlled mode is mainly caused by the vibration of transformer windings.

62

Figure 3.17. Comparison of the winding-controlled modes at (a) 229 Hz, (b) 420 Hz, (c) 533 Hz, and (d) 683 Hz in both radial and axial directions.

Figure 3.17 shows the experimentally and numerically calculated mode shapes

of the winding-controlled modes in both axial and radial directions. The measurement

points on the winding surface can be found in Figure 3.10, which correspond to the

discrete points in Figure 3.17. The four columns of test points from left to right are the

left, back, front, and right sides of the transformer winding. The four vertical lines

represent the undeformed winding and are overlapped in the same figures for

comparison. As can be seen in Table 3.2, the FE calculations of the winding-controlled

modes agree well with the experimental measurements. For the first four winding-

controlled modes, the maximum frequency deviation is 14.4%, while the deviations in

other modes are less than 5.0%.

The transformer winding assembly is wound in advance and stacked into the

core limbs as a whole. This heavy mass of winding assembly is then supported by the

bottom yoke. An insulation layer made of wood or resin materials is normally inserted

in between. By doing so, the sub-system comprising the winding, insulation layer, and

(a) (c)

(b) (d)

63

bottom yoke can be regarded as a mass–spring system, where the winding assembly is

the lumped mass and the insulation layer acts as a spring. As a result, there will be a few

winding-controlled modes related to how the winding is mounted on the bottom yoke.

The measured mode at 229 Hz is one of these, and is a rigid-body translation mode

along the axial direction. The other boundary-relevant modes will be introduced in the

third category as coupled modes.

The 2nd winding-controlled mode in Figure 3.17(b) is a (2, 1) cylindrical mode

with both ends constrained while the 4th winding-controlled mode in Figure 3.17(d) is a

(2, 2) cylindrical mode. These are defined by the nodal line numbers in the longitudinal

and circumferential directions. These two modes reveal that the winding vibration is

most likely to be similar to a cylindrical shell. Thus, the winding assembly cannot be

treated as a series of lumped masses or ring stacks.

The 3rd mode is a combination of winding bending modes in both axial and

radial directions, which deform the transformer winding in two directions

simultaneously. Although they were not observed in the test frequency ranges, it is

expected that the higher order winding-controlled modes might involve the bending

modes of each disk. Excessive vibration under this mode will induce winding buckling

and plastic deformations.

(c) Core–winding coupled mode

The coupled modes relate to the interaction between the transformer core and winding

assemblies. The transformer core and winding are mechanically connected by the

bottom pressboard, which determines the support condition of the winding assembly.

Figure 3.18 shows the first four coupled mode shapes of the transformer, where modal

participation is not just generated by the core or by the winding assembly.

64

Figure 3.18. Comparison of the core-winding coupled modes at (a) 11 Hz, (b) 44 Hz, (c) 57 Hz, and (d) 154 Hz.

For example, in the 1st coupled mode, the transformer structure is bending back

and forth around the bottom yoke. Similarly, in the 2nd mode, the transformer structure

vibrates in-plane where the coupled parts move in-phase. Unlike the 2nd mode, the 4th

mode involves the winding and core in-plane vibration, while the coupled parts move

anti-phase. These three modes are all related to the supporting boundaries, which are

determined by how the transformer is mounted. The 3rd mode is mostly a rigid-body

vibration of the winding with respect to the core assembly. Vibration modes in these

cases are related to the supporting conditions of how the winding assembly is fixed on

the core yoke. Theoretically, these modes exist for all core-type transformers, no matter

how different they are in their design, manufacture, and on-site installation

configurations. It is worth emphasising that the resonances related to the coupled modes

are of great importance since they usually occur in a low-frequency range. The design

of vibration isolators or dynamic absorbers should account for these coupled modes.

(a) (b)

(c) (d)

65

However, recent research shows that the same type of transformer with identical design,

materials, and operating conditions can still exhibit apparently different vibration levels

[66]. This might be partly explained by the difference in supporting boundaries during

transformer installation. As well as their influence on transformer vibration and noise

emission, some winding structure failures have similar deformation patterns for certain

structure modes. Winding tilting as introduced in Figure 3.20 is one example, which

was observed in a short-circuit test [66]. Structural resonance frequencies may shift with

the gradual degradation of the insulation materials or changes in supporting conditions.

If one of the shifted natural frequencies coincides with the excitation frequency of the

force, then large vibration with a resonance mode may occur at the resonance frequency.

In this case, the breakdown of an operating transformer in the steady-state with trivial

disturbance is then anticipated.

3.4.3 Numerical simulation of transformer frequency response

In this simulation, the same FE model with a forced-free boundary condition is adopted

for the steady-state dynamic response calculation in the FE software. In order to

facilitate experimental verification, a unit point force was imposed at the D1 position

(see Figure 3.10) along the normal direction. This was kept the same as in the modal

test. The comparison of the spatially averaged FRFs is presented in Figure 3.19, where a

good agreement can be found for such a complex structure. Combined with the modal

analysis, it is possible to infer that the vibration modelling of the power transformer is

feasible based on the FE method. In other words, the study of transformer vibration with

various structural faults, which would be too costly for experimental investigation, can

be simulated in the FE model. The following section investigates transformer vibration

when the winding undergoes global movement or local deformations.

66

Figure 3.19. Comparison of the FRFs between FE and impact test results.

3.5 Simulation of transformer vibration with winding damage

Excessive vibration of the transformer core will likely cause degradation of the SiFe

insulation coating. The transformer core may become overheated once the insulation is

locally damaged. As a result, a higher core loss, serious oil degradation, and even

transformer failure might occur. However, few core faults have been reported in

previous literature. In most cases, mechanical damage of the core is commonly found to

result from clamping looseness. Compared to core failure, winding anomalies in terms

of global movements and local deformations have been extensively observed in recent

decades. The impact of a short-circuit impact or magnetising inrush current could

generate a large additional mechanical burden in the winding, and thus affect the

mechanical integrity of the transformer. The side effects, such as loss of winding

clamping pressure and damage to the insulation design, lead to insulation deterioration

and finally the breakdown of the transformer. This section discusses the vibration

100 200 300 400 500-50

-40

-30

-20

-10

0

Frequency [Hz]

FRF

[dB]

TestFEM

67

characteristics of the transformer winding in the presence of winding deformations and

movements.

The types of winding damage under investigation and the associated parameters

are shown in Figure 3.20, where winding movement and local deformation are included

individually in each case study.

HV Elongation d = 10 mm

LV Buckling d = 10 mm

Winding Tilting θ = 0.0226 rad

Winding Twist θ = 0.0718 rad

Figure 3.20. Schematics of types of winding damage introduced to the FE model.

For elongation of the HV winding and buckling of the LV winding, a 10 mm

displacement is introduced on the front side. The reason for choosing elongation of the

HV winding and buckling of the LV winding is related to the EM forces they

experience. As introduced in Chapter 2, the HV winding suffers from a compressive

EM force, while the LV winding suffers a tensional force. For winding movement,

68

global tilting and twist are studied as they have been found in transformer accidents.

The tilt and twist angles are θ = 0.0226 rad and θ = 0.0718 rad, respectively. The

variations of natural frequencies and mode shapes of the winding assembly, and the

related vibration energy distribution, are illustrated and explained based on the FE

modal analysis.

Table 3.3. Natural frequency shifts of the winding-controlled modes due to winding deformations (Hz).

Mode order Normal HV Elongation LV Buckling Winding Tilting Winding Twist maxnf

1 267.71 266.95 267.71 267.66 267.65 −0.76

2 415.30 408.17 416.01 415.26 415.19 −7.13

3 554.81 566.80 539.57 554.76 554.39 −15.24

4 716.62 703.74 709.61 716.38 718.18 −12.88

As can be seen in Table 3.3, the natural frequencies of the winding-controlled

modes change in the presence of winding damage. The maximum deviations of the

natural frequency are all negative in the investigated modes. However, the elongation of

the HV winding increases the 3rd natural frequency up to 12 Hz, which denotes that the

natural frequency shift is not always negative or positive when damaged.

As well as the frequency shifts, the distributions of the corresponding modes are

also changed. Figure 3.21 shows a comparison of modal shapes between normal and

damaged windings. With global winding tilting, the peak vibration energy shifts slightly

to the right side, although the energy distribution pattern does not noticeably change. It

is shown in Figure 3.21 (2nd mode) that the contoured modal displacement is almost

symmetrical in the normal condition, whereas its distribution becomes asymmetrical

when tilted to the right. With respect to the 3rd mode, winding twisting causes a

rearrangement of the nodal lines, which are rotated in the opposite twist direction.

69

Figure 3. 21. Comparison of the modal shapes of normal and damaged windings (dot-dashed line marks the centre of the winding).

Theoretically, it is straightforward to use the modal displacement distribution as

a monitoring indicator. However, the measurement of winding modes is difficult in

operating condition, even when the transformer is off-line. The experimental technique

and possible on-line implementation will need further development in order to utilise

mode-shape-based monitoring techniques.

It should be mentioned that the cooling oil for oil-immersed transformers is able

to change the vibration response of a “dry” system, i.e., the resonance frequency and

mode shape functions, through fluid–structural (FS) coupling. When the transformer

winding and core are situated in a metal enclosure and immersed in the cooling oil,

vibration is transmitted to the enclosure via the FS and structural–structural (SS)

coupling. Assuming that winding tilting occurs, the winding vibration varies

accordingly and the vibration distribution of the enclosure will be changed as a result of

such couplings. Although the vibration transmission path is not changed in the presence

2nd mode

3rd mode

Normal

Normal

Tilting

Twisting

70

of winding tilting damage, the winding tilting introduced in Figure 3.20 is prone to

increase the vibration response of the right side of the enclosure. By comparing the

vibration energy distributions of the transformer tank, it may be possible to successfully

monitor the mechanical conditions of the internal windings.

3.6 Conclusions

A numerical modelling based on the FE method was used to predict the vibration

response of core-form transformers consisting of winding and core assemblies. It was

verified by the EMA method that the numerical model, as well as the relevant model

simplifications, were acceptable, reliable, and advisable in general. Moreover, it was

shown that this modelling approach would be applicable to a broad category of

transformers owing to the portability of the FE method.

With this model, the vibration frequency response of a 10-kVA small-

distribution transformer was calculated. Good agreement with the experimental results

was found. Based on the 3D FE model, three types of structural anomalies in

transformer winding were simulated using the FE model: 1) local deformation, 2)

winding tilting, and 3) winding twisting.

During the EMA model verification, the modal characteristics of a core-form

power transformer were discussed thoroughly. The transformer vibration modes were

classified into winding-controlled, core-controlled, and winding/core-coupled modes.

This approach made the description of transformer vibration more specific. It was found

that the transformer modes were not always in the high-frequency range, despite the

transformer being mostly constructed from copper and steel materials with high

stiffness values. Experimental observations and numerical simulations both showed that

the low-frequency modes were usually related to the core-controlled and coupled modes.

71

It was also found that the transformer vibration could not be treated as a lumped

parameter system when discussing its dynamic responses. Instead, it became more like a

cylindrical structure with both ends constrained, as concluded from its modal analysis.

As an alternative method to factory testing, FE analysis provides a way to estimate the

dynamic response of a transformer under simulated structural damage.

72

Chapter 4 Mechanically and Electrically Excited Vibration

Frequency Response Functions

4.1 Introduction

In recent decades, efforts have been made to use a transformer’s vibration, voltage, and

current to detect possible changes in the electrical and mechanical properties of the

transformer [43]. Much of this previous work has examined “response-based”

monitoring methods, making use of only the transformer vibration signal to evaluate the

health status of the transformer. Berler et al. proposed a method based on an

experimental observation that transformer vibration increases when the clamping

pressure on the transformer winding is reduced [44]. Other vibration-based methods

mainly utilise relative parameters, extracted from either the frequency or the time

domain of a transformer’s vibration signals, to relate to certain transformer faults [45–

47]. However, these methods are only effective in detecting faults corresponding to the

transformers used in their case studies, and are difficult to apply to other transformers

with different designs or operating conditions.

On the other hand, the desire for more effective methods to detect transformer

faults has motivated the study of the physical causes of transformer vibration and its

changes in terms of the modal characteristics of the transformer structure and the

properties of the excitation forces. Henshell et al. calculated the natural frequencies and

modal shapes of a transformer core by considering the core as a framework of beams

[52]. Kubiak presented a vibration analysis of the transformer core under a normal no-

73

load condition with two accelerometers, and found a lower resonance frequency at

around 100 Hz [55]. More recently, Li et al. analysed the vibration of a transformer’s

pressboard, winding, and insulation blocks using the finite element (FE) method [67].

Michel and Darcherif conducted an experiment to identify the natural frequency of a

transformer using a shaking table [68]. Zheng et al. measured the spatial distribution of

the winding vibration of a distribution transformer (which is the same as that described

in this chapter) under electrical excitation [69]. In their study, a laser Doppler scanning

vibrometer was used to reveal detailed spatial characteristics of the transformer winding

with various degrees of mechanical faults.

It is well known that transformer vibration is usually generated by

electromagnetic force in the windings and magnetostrictive force in the core. Both

forces are spatially distributed and are unlike the point force used in traditional modal

testing. The differences between the frequency response functions (FRF) of the

transformer structure due to point force excitations and those due to distributed

electromagnetic and magnetostrictive excitations has not yet been studied.

Understanding this difference is important when the FRF of the transformer vibration is

used for detecting possible damage in a transformer structure.

Along this direction of scientific research, in 2007, Phway and Moses observed

the magneto-mechanical resonance of a single-sheet sample [70]. Yao extended Phway

and Moses’ work to a three-phase transformer core in 2008 [71]. In the most recent

work, in 2012, Shao et al. presented their results in terms of the vibration FRF of a

power transformer under only electrical excitation [72]. There is an obvious need for a

study comparing the electrically and mechanically excited transformer vibration FRFs

in order to effectively utilise knowledge obtained in the fields of electrical and

mechanical engineering for vibration-based detection of transformer faults.

74

This chapter focusses on the FRFs of a 10-kVA distribution transformer due to

mechanical and electrical excitations. The mechanical excitation involves the use of an

impact hammer to excite the transformer at one position with an impulsive force, while

the electrical excitation employs a sinusoidal voltage at the primary winding input. The

modal parameters are identified in a modal test and compared between the two

excitation methods. A comparison of the modal properties of the transformer found

from the two excitations reveals the differences in the amplitudes of the frequency

responses. However, the natural frequencies of the transformer determined from the two

experiments remain unchanged.

4.2 Methodology

In the traditional modal test, test structures are typically excited with one or more point

excitation sources, and the responses are measured at a few locations on the structures.

Modal parameters can be extracted from the FRFs of the response signals to the input

signals. Impulse sources are also used to produce the impulse response functions (IRF)

of the structures. The FRF and IRF are related via Fourier transform pairs. Although the

IRF here is simply for the calculation of the corresponding FRF, the structural features

of the transformer may also be extracted from the time-domain rising and settling times

and the peak vibration value. A typical example of this is the use of the features of the

transient vibration of a power transformer due to energisation to detect the winding

looseness [73].

For point-force excitations, the vibration response at location ix of the structure

(as an output) is related to the point forces (as inputs) as:

'( | ) ( , | ) ( | )i M i k kk

v x H x x F x , (4.1)

75

where ( | , )iM kH x x is the mechanical FRF between ix and kx , and ( )kF x is the point

force at location kx . For distributed force excitations, the vibration response at the same

location of the structure is expressed as:

ˆ( | ) ( , | ) ( | )i iM k k kV

v x H x x F x dx , (4.2)

where ˆ ( )kF x is the force per unit volume at location kx and V is the entire volume of

the transformer structure.

To measure the electrically excited FRF of the transformer, a sinusoidal voltage

input is applied at the primary winding. Unlike the point force input, the distributed

force in the transformer winding and core cannot be measured. The measurable input for

this case is the primary voltage, ( ) ( )cos( )o o o oU U t , where o is the testing

frequency. It is worth noting that the distributed force is also related to the winding

current. To simplify this first experiment, the secondary winding is in an open circuit

condition. As a result, the current-induced electromagnetic force in the winding is

negligible.

Because the electrical inputs used in the experiment are much smaller than the

saturation values of the test transformer, it is therefore reasonable to approximate the

magnetostriction in the transformer core by the square of the magnetic flux density

( )o [74], which is in turn proportional to the voltage as [75]:

( )( )2.22

oo

o

UNS

, (4.3)

where N is the number of turns of the primary winding and S is the cross-sectional

area of the core. A more detailed nonlinear relationship should be used to describe the

dependence of the magnetostriction on the flux density if the transformer is operated

close to the deeply saturated region. Nevertheless, Eq. (4.3) allows for selection of the

76

applied voltage at various frequencies such that the flux is at a non-saturated value (0.91

T). A test frequency range from 15 Hz to 60 Hz is selected based on the requirements

for the signal-to-noise ratio and the limitations of the voltage input. Therefore, vibration

from 30 Hz to 120 Hz can be measured, owing to the quadratic relationship between the

flux and the magnetostriction. At the frequency ( 2 o ) of transformer vibration, the

FRF between the body force and the input voltage can be expressed by:

2( )ˆ ( | ) ( | ) o o

Ek ko

UF x H x

, (4.4)

where ( | )E kH x is the electrical FRF between the voltage input and the transformer

body force at location kx (when the secondary winding is in the open circuit condition).

Using Eqs. (4.2) and (4.4), the vibration at kx and the primary voltage input are related

by:

2( )( | ) ( | ) o o

i iMEo

Uv x H x

, (4.5)

where the electrically excited FRF:

( | ) ( , | ) ( | )i iME M Ek k kV

H x H x x H x dx (4.6)

includes contributions from both the mechanically and electrically excited FRFs. On the

other hand, if the transformer is only excited by a single point force at kx , then only the

mechanically excited FRF is produced (see Eq. (4.1)).

4.3 Description of experiments

The test transformer was a 10-kVA single-phase transformer with rating voltages of

415/240 V. A detailed description of the transformer specifications can be found in a

final-year thesis [76] and have been briefly summarised in Chapter 3.

77

The actual experimental setup for obtaining the electrically excited FRF is

presented in Figure 4.1. A sinusoidal voltage signal from a signal generator (Agilent,

33120A) was amplified via a power amplifier (Yamaha, P2500S), and then a variac. As

a result, a 200-V voltage at each test frequency was applied to the primary input of the

model transformer. The transformer vibrations at 48 measurement locations were

measured using accelerometers (IMI, 320A). The outputs of the accelerometers were

pre-amplified using a signal-conditioning device before being sent to a laptop computer

for post-processing via a DAQ (NI, USB-6259).

Figure 4.1. The actual experimental setup for obtaining the electrically excited FRFs.

The experimental setup for obtaining the mechanically excited FRF consists of

the same model transformer and accelerometers shown in Figure 4.1. For this case, a

multiple-input system (B&K, Pulse, 3560) was used for data acquisition and calculating

FRFs at the same 48 measurement locations. The transformer vibration was excited

using an impact hammer (B&K, 8206). Only out-of-plane (normal to the surface)

vibrations were measured at all measurement locations for both the mechanically and

electrically excited cases. The impact hammer and accelerometers were calibrated

before each experiment.

78

The measurement points and excitation locations were identical to what was

described in Section 3.4.1. The reasons for selection of these points and the reciprocity

test are also the same. In order to avoid redundancy, they will not be repeated here.

4.4 Results and discussion

4.4.1 FRF due to mechanical excitation

Figure 4.2 shows the spatially averaged FRF. Four resonance peaks can be found

between 20 Hz and 120 Hz. To understand this averaged FRF, the FRFs at a few

representative test points are selected for detailed analysis.

Figure 4.2. Spatially averaged FRF of the distribution transformer subject to a mechanical excitation.

(1) FRF at the test point T01

This point is located on the left end of the top yoke, and is closest to the driving point.

The Bode diagram of this point is presented in Figure 4.3. Similar to the spatially

averaged FRF, four resonance peaks are found at the same frequencies in the

amplitude–frequency diagram. With respect to the 1st resonance peak, at 36 Hz, a 180°-

20 40 60 80 100 120-20

-15

-10

-5

0

Frequency [Hz]

FR

F [d

B]

79

phase change between 30 Hz and 40 Hz can be seen in the phase–frequency curve.

Damping at this frequency is relatively low.

Figure 4.3. Bode diagrams of the mechanically excited FRF at test point T01.

(2) FRF at the test point T40

The test point T40, located in the middle of the winding, is selected for analysis of the

resonance peak at approximately 53 Hz. Likewise, the Bode diagram is analysed in

Figure 4.6. A 180°-phase change is recorded in the Bode diagram corresponding to the

peak in the amplitude–frequency curve.

Figure 4.4. Bode diagrams of the mechanically excited FRF at test point T40.

20 40 60 80 100 120-30

-15

010

Am

plitu

de [d

B]

20 40 60 80 100 120-200

0

200

Frequency [Hz]

Pha

se [

o ]

20 40 60 80 100 120-30

-15

010

Am

plitu

de [d

B]

20 40 60 80 100 120-200

0

200

Frequency [Hz]

Pha

se [o ]

80

(3) FRFs at the test points T25 and T33

The same analysis was undertaken for T25 and T33, on the left and right sides of the

core structure, respectively (see Figure 4.1). After analysing the Bode diagram in Figure

4.5, the 3rd and 4th natural frequencies are verified to be 77 Hz and 103 Hz, respectively.

Figure 4.5. Bode diagrams of the mechanically excited FRF at test points T25 and T33.

(4) Mode shapes at the resonance frequencies

Based on the frequency response at all the test points, the mode shapes are sketched as

shown in Figure 4.6. As can be seen in Figure 4.6, the 1st mode is actually a torsional

mode. The top yoke rotates around the central limb and the side limbs are vibrating out

of phase. Since the bottom yoke is constrained by a support, its response is relatively

small compared with the top yoke. The 2nd mode at 53 Hz is mainly a result of the

rocking vibration of the transformer winding. From the mode shape, it can be seen that

this mode causes the winding assembly to bend backwards and forwards. The 3rd mode

and 4th mode are the bending modes with respect to the left and right limbs of the core.

These resonances (except for mode 2) are clearly not from rigid-body modes, because

their corresponding mode shapes demonstrate bending deformation in the limbs of the

core structure, which are the typical characteristics of structural transverse modes. The

low resonance frequencies of such a transformer core are due to the relatively small

20 40 60 80 100 120-30

-20

-10

0

Am

plitu

de [d

B]

20 40 60 80 100 120-200

-100

0

100

200

Frequency [Hz]

Pha

se [

o ]

20 40 60 80 100 120-30

-20

-10

0

Am

plitu

de [d

B]

20 40 60 80 100 120-200

-100

0

100

200

Frequency [Hz]

Pha

se [

o]

81

stiffness of the core in the transverse direction (perpendicular to the core lamination). A

similar observation of the core structural modes in the transverse direction for hundreds

of hertz was also reported in previous research on a transformer of similar size [55].

Mode 1 (35 Hz) Mode 2 (53 Hz)

Mode 3 (77 Hz) Mode 4 (103 Hz)

Figure 4.6. Mode shapes at the corresponding resonance frequencies.

Although the structure appears to be symmetrical in terms of geometry, the

effective local Young’s modulus and shear modulus may be asymmetrical at both side

limbs because of the asymmetrical locations of the joints between the silicon-iron (SiFe)

sheets and because of the composition of the sheets. The results indicate that the left

limb is a little bit “softer” than the right limb. This is why the natural frequency of the

left-limb bending mode is lower than that of the right-limb bending mode.

82

(5) Predicted modal characteristics

An FE analysis was conducted to validate the experimental observation. The FE model

takes into account the anisotropic properties of the transformer core and winding

assembly. The predicted natural frequencies and corresponding mode shapes are

presented in Figure 4.7. A good agreement is found between the predicted and measured

results. The above analysis also demonstrates the potential of using mathematical tools

for modelling the transformer’s FRFs. As indicated in Ref. [75], an analytical method

based on the Euler–Bernoulli beam theory was adopted to estimate the natural

frequency of the 1st bending mode, which was calculated as 31.1 Hz considering the

anisotropic Young’s modulus of the core assembly. As well as validating the

experimental observation, this calculation also implies the dependence of the

transformer resonance on the core stiffness.

Mode 1 (36.7 Hz)

Mode 2 (48.3 Hz)

Mode 3 (82.3 Hz)

Mode 4 (115.8 Hz)

Figure 4.7. Predicted natural frequencies and mode shapes of the model transformer.

83

Although the identified first four modes of the model transformer have natural

frequencies below 120 Hz, the mode shapes of these modes indicate that they consist of

both rigid-body vibration (which is highly dependent on the supporting and boundary

conditions) and vibration distributed in the core structure (which is dependent on the

local stiffness and mass density). As a result, these modes carry the general features of

the transformer structure and have several features in common in the mid- and high-

frequency ranges. It is also noted that the high-frequency modes of a transformer often

present a broadband frequency structure owing to the increased modal density and

damping. For these modes, the changes in the characteristics with the changes in

structural conditions may be difficult to detect individually.

(6) Effect of different boundary conditions

The active parts (core and winding) of all power transformers are supported at the

bottom by their own weight for structural stability. This work is to study the effect of

boundary constraints on the transformer’s FRFs, and to obtain an estimate of the

resulting changes in the FRFs when the boundary constraints vary from a well-clamped

boundary condition to a force-free boundary condition. To create a well-clamped

boundary condition, the top parts of the transformer core and winding were fixed to

very stiff and heavy steel beams, which in turn were connected to the building structure.

The natural frequencies with and without the clamping arrangement are compared in

Table 4.1.

In summary, the increased boundary constraints at the tops of the active parts

caused an overall increase in the natural frequencies, indicating an increase in the

overall structural stiffness. The low frequency modes are affected the most, with a

maximum 40% deviation in natural frequency. Furthermore, as expected, the clamped

84

boundary condition suppresses the vibration response at the top yoke and winding,

which contributes significantly to the modal response of the 1st and 2nd modes.

Table 4.1. Comparison of the natural frequencies of the model transformer under supported-clamped and supported-free boundary conditions.

Mode order 1st 2nd 3rd 4th

Supported-free (Hz) 35 53 77 103

Supported-clamped (Hz) 49 69 82 105

Deviation (%) 40.0 30.2 6.5 1.9

4.4.2 FRF due to electrical excitation

The spatially averaged FRF due to electrical excitation is presented in Figure 4.8, where

the four mechanical resonance peaks are readily identifiable. Referring to Eq. (4.6), the

electrically excited FRF ( ( | )E kH x ) also contributes to the FRF in series. If ( | )E kH x

had any resonances in the frequency range of interest, then the corresponding FRF

would have shown resonance peaks in Figure 4.8. The resonances of ( | )E kH x are

determined by the distributed inductance and capacitance of the transformer winding.

The electrical FRF of the model transformer showed that its first resonance frequency

occurred at 2.15×106 Hz.

85

Figure 4.8. The spatially averaged FRF of the transformer vibration due to electrical excitation.

This result demonstrates that (1) mechanical resonances in a transformer can

also be excited by the distributed magnetostrictive excitation and (2) there are no

electrical resonances in the frequency range of interest in this investigation.

Since the force of the electrical excitation is unknown, the absolute values of

these two FRFs cannot be compared directly. However, the relative value of each test

can be analysed. A comparison of the FRFs in Figures 4.4 and 4.8 shows that the

mechanically excited FRF has a maximum response at 36 Hz, while the electrically

excited FRF has a maximum response at 54 Hz.

Table 4.2. Level differences of the 2nd, 3rd, and 4th peak responses with respect to the 1st peak response.

Excitation V54 Hz – V36 Hz (dB) V78 Hz – V36 Hz (dB) V104 Hz – V36 Hz (dB)

Mechanical −9.47 −8.85 −8.90

Electrical 1.69 −2.85 −2.15

20 40 60 80 100 120-20

-15

-10

-5

0

Frequency [Hz]

FR

F [d

B]

86

For each case of excitation, the level differences of the peak responses of the 2nd,

3rd, and 4th modes with respect to the peak response of the 1st mode are listed in Table

4.2. The reduced peak amplitudes in the mechanically excited FRF can be readily

explained by the mechanical mobility function of a spring–mass–damper oscillator,

which is inversely proportional to the damping constant. The well-known phenomenon

of increased structural modal damping with an increase in natural frequencies

corresponds qualitatively to the reduced peak amplitude in this case. However, the peak

response of the electrically excited FRF, MEH (see Eq. (4.6)) is not only inversely

proportional to the modal damping constant, but also directly proportional to EH (see

Eq. (4.4)). The increased trend in the peak responses of MEH suggests that EH may

increase with frequency. However, further analysis is required to explain the physical

mechanisms involved.

As magnetostriction is a nonlinear function of the flux, it is necessary to

examine experimentally the changes of the electrically excited FRF for different flux

inputs. The spatially averaged FRFs for three applied flux values are shown in Figure

4.9. Although the amplitude of the FRF increases with the increase in flux density, the

natural frequencies remain the same. When the flux density is 1.13 T, the transformer

core becomes closer to the saturated region, which is evidenced by the magnetising

hysteresis loops in Figure 4.10. In this test, a 0.79-T flux excitation is sufficient to

display the modal characteristics of the transformer structure.

87

Figure 4.9. Spatially averaged FRFs of the transformer vibration due to electrical

excitation at different RMS flux densities.

The dependence of the electrically excited FRF on the applied flux density

implies that the square relationship between the magnetostrictive body force and the

flux density, given by Eq. (4.4)), is inadequate. Although a more accurate nonlinear

expression for the body force can be established based on the experimental data shown

in Figure 4.9, the study and understanding of the physical mechanisms associated with a

correct nonlinear expression would be a significant future work.

Figure 4.10. Magnetising curves of the model transformer at five different frequencies.

20 40 60 80 100 120-20

-15

-10

-5

0

5

Frequency [Hz]

FR

F [d

B]

0.79T0.91T1.13T

-0.4 -0.2 0 0.2 0.4-2

-1

0

1

2

Magnetizing Current [A]

Flux

Den

sity

[T]

15Hz25Hz35Hz45Hz55Hz

88

Figure 4.10 also demonstrates that, although the root-mean-square (RMS) flux

density was kept at 1.13 T, the hysteresis loop is not the same at different frequencies.

Greater hysteresis effects and energy losses are found for higher frequency excitations.

A direct application of these measurements is to detect transformer faults off-

line via the swept-sine method. In practice, the vibration modes relative to the active

parts of the transformer are difficult to excite using mechanical excitation. More likely,

the modes of the transformer enclosure will be detected in such cases. The FRFs from

mechanical excitation are generally not able to reflect transformer faults. However,

electrical excitation from inside the transformer would be able to overcome this

shortcoming. As reported in this chapter, the natural frequencies of the active parts of

the transformer can be identified from the electrical excitation. The occurrence of a

structural fault, e.g., a winding deformation, will change the FRFs and will eventually

be detected following comparisons.

The results presented also suggest an on-line application for transformer

condition monitoring based on FRF analysis. To examine the health condition of a

practical power transformer, the values of the FRF can only be measured at discrete

frequencies, such as 100 Hz and its harmonics. If the transformer is affected by a slowly

varying magnetic field, such as a geomagnetic field, then an offset in the operating

magnetic flux density in the core may generate FRF values at 50 Hz and its harmonics.

If the mechanical properties (such as the clamping pressure or stiffness of the insulation

material) of the transformer change, then the resonance frequencies and the shape of the

FRF of the transformer will also change. Such a change is also expected to be seen at

those discrete frequencies. Indeed, the detection of structural faults is only dependent on

the changes in the frequency response itself. Even if the excitation frequencies are not

very close to the natural frequencies of the transformer FRF, the structural faults will

89

still cause some changes in the response at those off-resonance frequencies. These

changes may not be as obvious as those at the natural frequencies.

However, transformers are often subjected to excitations with a broadband

frequency. Further work is underway to measure the electrically excited FRFs caused by

the transient excitations during transformer energisation and by random excitations

caused by variation of the loading. If one of the resonance frequencies is close to one of

those discrete “excitation” frequencies, then the change in the FRF due to structural

faults at this discrete frequency would be more pronounced. Therefore, identifying the

resonances close to the excitation frequency and observing the changes in the FRF at

the excitation frequencies may hold the key to an empirical estimation of the modal

properties of the transformer, which may in turn be related to the properties of the

transformer faults.

4.4.3 Effects of different clamping conditions

Additional case studies were undertaken to examine the variation of the mechanically

excited FRF of the model transformer to the changes in the clamping force of the core.

Figure 4.11 shows the spatially averaged FRFs of the transformer with the normal

clamping condition, 50% clamping force on the left core limb, and the same on the right

core limb. The FRFs with loose core clamping force differ significantly from those with

normal clamping force. With 50% looseness at the left limb, the natural frequency of the

3rd mode is reduced by 7.8%, while with 50% core looseness at the right limb, the

natural frequency of the 4th mode is decreased by 7.7%. The same result was also

observed in the electrically excited FRF, which is presented in Figure 4.11(b). A

comparison between these three cases indicates that the change in structural stiffness

induced by the core-clamped force can be readily detected by the mechanically excited

FRF as well as the electrically excited FRF.

90

Figure 4.11. Spatially averaged FRFs of the transformer vibration with clamping

looseness under (a) mechanical and (b) electrical excitations.

A comparison between these three cases indicates that the change in structural

stiffness induced by the core-clamped force can be clearly observed in the mechanically

excited FRF. The changes in the different modal frequencies with respect to different

locations of core looseness also suggest the possibility of using this knowledge to

determine the location of the looseness. Similar observations were also made on the

electrically excited FRFs.

20 40 60 80 100 120-20

-15

-10

-5

0

Frequency [Hz]

FRF

[dB

]

Normal case50% looseness at left limb50% looseness at right limb

20 40 60 80 100 120-20

-15

-10

-5

0

Frequency [Hz]

FR

F [d

B]

Normal case50% looseness at left limb50% looseness at right limb

(a)

(b)

91

4.4.4 FRFs of a 110 kV/50 MVA 3-phase power transformer

The resonance characteristics in the FRF are in fact general properties of all practical

power transformers. The mechanically excited FRF of a 110 kV/50 MVA power

transformer is used to support the above statement. Shown in Figure 4.12 is the spatially

averaged vibration of the core and windings of the transformer. The excitation location

was at the left end of the top yoke. Compared with the FRFs of the model transformer,

the FRFs of the 110 kV/50 MVA power transformer demonstrate a more complex

frequency distribution. However, the resonance peaks below 600 Hz are clearly

observable.

Figure 4.12. The mechanically excited FRFs at (a) the core and (b) the winding of a

110 kV/50 MVA power transformer.

4.5 Conclusions

This work presented the first experimental comparison between the mechanically and

electrically excited FRFs of a small-distribution transformer. The comparison

demonstrated that both mechanically and electrically excited FRFs carry information

about the modal characteristics of the transformer structure. This result has not only

92

academic value, but also practical significance because the modal characteristics of a

transformer structure are related to the causes of a transformer’s mechanical faults.

Experimental evidence was also provided in this chapter to show that the causes of

transformer faults (such as a reduction in core clamping force) change the resonances in

the measured FRFs. The difference between mechanically and electrically excited FRFs

was also explained. In particular, the effects of hysteresis of the core material on the

electrically excited FRFs under different levels of voltage excitation were discussed.

Although the above results were obtained from studying a small-distribution

transformer, the methods for extracting the modal characteristics from mechanically and

electrically excited FRFs and identifying the causes of mechanical faults using the

measured modal characteristics can be applied to all types of transformers. This is

because the vibration properties of transformers of different sizes are all controlled by

the same mechanical principles. The resonance phenomena of the transverse modes in

small-distribution transformers and the explanation of these add value to the traditional

understanding of the frequency range of resonances in transformer core vibration.

Although practical transformers vary widely in their mechanical and electrical

structures and operating details (e.g., loading characteristics and location of the voltage

tap-changer), the key features of the FRFs and the differences between the FRFs under

different excitations still apply. They provide a useful understanding of the modal

response of a transformer when condition monitoring techniques based on vibration

response are used for mechanical fault detection in a transformer. The vibration FRF

excited by an electrical current and FE modelling of the vibration modes of a 50 MVA

transformer will be a topic for future work

.

93

Chapter 5 Changes in the Vibration Response of a Transformer

with Faults

5.1 Introduction

In the power industry, the monitoring of health conditions and the detection of the

causes of failures of power transformers are often implemented using one of three

methods: dissolved gas analysis (DGA), frequency response analysis (FRA), and

vibration-based methods [45, 77, and 78]. These methods focus on measuring the

indicators of transformer faults and correlating the trends of changes in these indicators

with respect to the causes of transformer failure. DGA evaluates the transformer’s

health status by sampling and examining cooling oil, where changes in gas

concentrations, generation rates, and total combustible gases (TCG) are often used as

failure indicators [79]. Different interpretation schemes, based on empirical assumptions

and practical knowledge, have been developed for DGA to classify the features of

certain transformer faults. For example, an increase in TCG is correlated with possible

thermal, electrical, or corona faults in the transformer. FRA is concerned with changes

in the electrical frequency response within a critical high-frequency range [80], as faults

in the winding can cause changes in the distributed capacitance and inductance of the

winding.

As an online and nonintrusive method, vibration-based condition monitoring of

a power transformer’s health status has attracted considerable attention in recent

decades. Previous work demonstrated that this method provides an option for assessing

94

the mechanical integrity of a transformer [44, 46, 47, and 81]. The vibration-based

condition monitoring method also relies on changes in the vibration response of the

transformer. It focusses on spectrum analysis of the vibration at twice the operating

frequency and its harmonics, where looseness in the winding clamping force may cause

variations at those frequencies [44]. The transient vibration triggered by transformer

energising/de-energising operations has also been employed to detect abnormalities in

transformer winding [81].

Although the feasibility of using the vibration method for transformer condition

monitoring has been verified in these case studies, there is still a need to understand the

physical correlation between the changes in the frequency spectrum and changes in the

transformer’s mechanical properties associated with the causes of failure. Efforts have

also been made in the area of signal processing in order to extract the vibration features

of a damaged transformer by advanced signal processing methods, including the

wavelet transform, the Hilbert Huang transform, and their combinations [82]. This

approach also requires a link between the extracted features and the physical

mechanisms causing them. Since the current state-of-the-art in vibration-based

condition monitoring of transformers also relies on the analysis of changes in

transformer models [3, 4], it becomes clear that a study of the sensitivity of the

transformer’s vibration response to various causes of failure is necessary.

The vibration response of a power transformer is a measure of the transformer

vibration (as outputs) with respect to the transformer’s electrical inputs. Because the

vibration of the winding and core are nonlinear functions of the electrical inputs, the

traditional concept of the frequency response function (FRF) for linear systems does not

apply. In a previous work [83], the vibration response of a small-distribution

transformer to a sinusoidal voltage input with fixed amplitude was examined. It was

95

found that the steady-state response is characterised by the frequency components at

twice the excitation frequency and its harmonics. Therefore, the nonlinear vibration

response of the transformer with respect to a sinusoidal input may still be specifically

defined in the frequency domain. For example, if the secondary winding is in an open

circuit condition, then the vibration response function is defined as:

21

( | ) ( | 2 )i k i ok

H x H x k

, (5.1)

where ix is the measurement location of the vibration response, o is the excitation

frequency, and the frequency response component:

2

2( | 2 )

( | )( )

ok k o

i

o

kH x k

v xU

is defined by the ratio of the vibration component 2( | )oi kv x at 2 ok and the primary

voltage amplitude at o . This definition of the input and output relationship of a

transformer is practically significant as almost all in-service power transformers are

excited by a sinusoidal voltage. For many practical applications, a large percentage of

the vibration energy is contained at 2 o . Therefore, the first term of the response

function 2( | ) ( | 2 )i i oH x H x was used as the first-order approximation of the

vibration FRF of the transformer.

This chapter is an extension of the previous work in Chapter 4 on mechanically

and electrically excited FRFs of a small-distribution transformer. It focusses on the

sensitivity of the FRFs of a model transformer to the causes of faults. Experimental

evidence is presented to give a quantitative description of the causes of artificial faults

and to extract features of variations of FRFs that might be useful to the vibration-based

detection of the causes of transformer faults in general.

96

5.2 Theoretical background

The vibration response at location ix with respect to a distributed force excitation in

the transformer mechanical system can be expressed as [83]:

0ˆ( | ) ( , | , ) ( | )i M i k k k

V

Mv x H x x F x dx , (5.2)

where ( , | , )M i k MH x x is the mechanical FRF between ix and jx , and ˆ ( | )kF x is

the force per unit volume at location kx , and V is the entire volume of the transformer

structure. Unlike the traditional definition of an FRF, a mechanical parameter vector

1 2[ , ,..., ]M M M MP is used to describe the causes of a transformer’s faults.

Following the same logic, the distributed force can also be described by an electrical

FRF that relates ˆ ( | )kF x as outputs and the sinusoidal voltage ( )oU as an input. As a

result of this analysis, the first-order approximation of the vibration FRF of the

transformer (as described in Eq. (5.1)) is expressed as:

)( | , , ( , | , ) , )( |i M E M i k k E k

V

M EH x H x x H x dx , (5.3)

where 1 2[ , ,..., ]E E E EQ is an electrical parameter vector. How to relate the

parameters in the mechanical and electrical parameter vectors to the causes of

transformer failures and what is the sensitivity of H with respect to the changes in M

and E are the challenging questions for vibration-based fault diagnosis of power

transformers. The variation of H with respect to the system parameters can be

expressed as:

1 1

( | , , )QP

i M E Mp Eqp qMp Eq

H HH x

. (5.4)

For the electrically excited FRF, which is mostly relevant to the input/output

relationship of an in-service transformer, the variation of the mechanically excited FRF

97

of the small-distribution transformer with only the mechanical parameters can be

evaluated. It should be noted here that the vibration-based method for detecting the

causes of faults is mainly interested in those causes related to mechanical parameters.

Although changes in the electrical parameters will also cause variation in the electrical

FRF of the transformer, such variations often occur in a much higher frequency range

and other techniques such as FRA have been developed for such detection. If the

transformer is excited by a point force at ox , then this mechanically excited FRF can be

expressed via a volume integration of ( , | , )M i o MH x x , as shown in Eq. (5.2). Its

variation with the mechanical parameter vector is:

1

( , | , )P

i o Mpp Mp

MM

MH

H x x

. (5.5)

The relationship between variation of the mechanically excited FRF (Eq. (5.5)) and that

of the electrically excited FRF (Eq. (5.4)) can be found by expanding the first term on

the right-hand side of Eq. (5.4):

1 1

[ , )] ( |P P

MMp Mp k E k

p pVMp Mp

EHH

H x dx

, (5.6)

indicating that the variation of )( | , ,i M EH x with respect to M is the spatially

averaged sensitivity of ( , | , )M i o MH x x over all the forcing locations and weighted

by , )( |k EEH x .

5.3 Description of experiments

The measurement of vibration response was performed on a 10-kVA single-phase

transformer with rating voltages of 415/240 V. The experimental set-ups were kept the

same as in Chapter 4, where the mechanical and electrical excitations were implemented

98

by an impact force and a swept-sine voltage, respectively. In order to stay consistent

with the previous study, the same impact location and test points were used.

To describe the causes of mechanical failures of a small-distribution transformer

using the mechanical parameter vector M , the percentage looseness of winding

clamping pressure and the percentage looseness of core clamping pressure are used as

two independent components in M . For example, to describe the changes in the core

clamping force, the first element in M is defined as (0)1 1 1(1 )M M , which changes

the nominal value of (0)1M (set by the manufacturer) to zero, where 10 1 is the

percentage looseness of the core. Similarly, the winding clamping force is described by

(0)2 2 2(1 )M M with 20 1 .

For this model of transformer, its winding and core segments are fastened by

four and eight bolts, respectively [83]. The nominal clamping forces set by Universal

Transformers are approximately 2250 N in the winding and 1500 N in the core; these

values are converted from the tightening torque by considering the bolt diameters,

thread lead angle, etc. [84]. The realisation of the aforementioned mechanical failures in

terms of clamping looseness depends upon the gradual adjustment of the preloading of

the corresponding clamping bolts using a torque wrench. Considering the importance of

diagnosing a fault in its early stages, a maximum 35% clamping looseness in the

winding and 25% looseness in the core were investigated with 5% increments in

both winding and core looseness.

In addition to the clamping looseness, the absence of longitudinal insulation

spacers was examined as another cause of mechanical faults, as these would reduce the

axial stability and cause excessive vibration. The design of longitudinal insulation and

arrangement of its mechanical faults, 3M , are presented in Figure 5.1, where eight

99

columns of insulation spacers are circumferentially equispaced along the winding circle.

The detailed dimensions of the insulation spacers and winding conductors can be found

in the partial view (A-A), where the height of the winding conductor and insulation

spacer are 8 mm and 3.2 mm, respectively. In addition to the thin dovetail spacers

between the winding disks, an array of 19.1-mm-thick insulation blocks are layered at

both ends of the winding assembly.

The fault of missing insulation spacers was introduced in the front column,

including the insulation blocks at both ends, as indicated in Figure 5.1. This column of

insulation spacers was separated into thirteen segments with approximately equal height,

corresponding to the thirteen missing insulation statuses of ( )3

nM shown in Figure 5.1.

The insulation spacers were removed cumulatively from 1n to 13n . All the

missing-insulation tests are conducted under the same winding clamping status.

Figure 5.1. The design of longitudinal insulation and the arrangement of missing

insulation spacers as a cause of mechanical faults.

5.4 Results and discussion

Generally speaking, the development of mechanical faults in a transformer structure is

accompanied by changes in the structural stiffness, mass, and damping. The resulting

100

variations in a transformer’s vibration responses due to some common faults will be

presented in the following subsections.

5.4.1 Vibration response due to core looseness


and (b) electrical excitations with core clamping looseness.

The first cause of mechanical failure under investigation is looseness of the

transformer core clamping force, which is described by (0)1 1 1(1 )M M with 10 1 .

By gradually reducing the clamping force with the same percentage increment

20 40 60 80 100 120 140-20

-10

0

10

20

30

Frequency [Hz]

FRF

[dB

]

1=20%

1=25%

1=15%

1=10%

1=5%

1=0

(a)

20 40 60 80 100 120 140-20

-15

-10

-5

0

5

10

15

20

25

Frequency [Hz]

FRF

[dB

]

(b)

1=25%

1=20%

1=15%

1=10%

1=5%

1=0

101

( 1 5% ), an overall 25% looseness was introduced to the left and right limbs

symmetrically. Based on the experimental methodology described in Chapter 4, the

vibration FRFs under mechanical and electrical excitations were measured. Spatially

averaged FRFs with different core clamping forces are presented in Figure 5.2, where a

5-dB offset from the FRF underneath is introduced for clarity. On each FRF curve, four

resonance peaks can be clearly discerned under electrical and mechanical excitations.

However, the resonance peaks at the 3rd and 4th mode responses are affected greatly in

the presence of core looseness, while the other modes seem unaffected.

A quantitative comparison of the changes in vibration at different clamping

statuses, by means of natural frequency shifts ( fn ) and cumulative changes in the

FRFs at 100 Hz ( 100MH Hz and 100H Hz

), can be found in Table 5.1. The

percentage natural frequency shifts were calculated around the lowest state ( 1 0 )

without core clamping looseness so as to give the cumulative changes of the frequency

responses at 100 Hz. Variations in vibration at discrete frequencies, i.e., 100 Hz and its

harmonics, are of most concern in response-based monitoring strategies. The reason for

selecting 100 Hz in the following analysis is that it is a forced-vibration frequency close

to the most affected mode at 107 Hz.

Table 5.1. Quantitative variation of the transformer vibration FRFs due to core clamping looseness.

2 (%) 5 10 15 20 25

nf (%)

1st mode −1.43 −2.85 −2.85 −4.28 −5.71

2nd mode −0.94 −0.94 −0.94 −0.94 0

3rd mode −1.3 −2.6 −3.25 −4.55 −5.19

4th mode −3.4 −4.37 −6.8 −8.74 −10.19

MH (dB) 100Hz 1.15 1.49 2.04 0.74 0.69

H (dB) 100Hz 0.51 0.91 1.77 2.56 2.12

102

From Table 5.1, a general decrease of the first four natural frequencies is

observed in the presence of the growing clamping looseness from 1 5% to 1 25% .

However, the frequency responses at 100 Hz increased dramatically under both

excitations. To facilitate the explanation of these variations, the mode shapes of each

resonance frequency are recalled in Figure 5.2(a), based on the previous reports in Ref.

[83]. According to the modal participation at each mode, the 1st, 3rd, and 4th modes can

be classified as core-controlled modes, which are dominated by the transformer core

assembly. The occurrence of clamping looseness directly affects the core-controlled

modes owing to the resulting stiffness reduction in the core assembly. With the

development of core looseness, the natural frequencies of the core-controlled modes are

consistently reduced within the tested looseness range. As can be seen from Table 5.1, a

maximum 10.19% (9.5 Hz) frequency shift was recorded at the 4th mode, which is of

great importance for vibration-based condition monitoring. Utilising the modal

parameter identification approach, the natural frequency shift can be detected and

related to possible structural damages.

In contrast, the 2nd mode appears to be unaffected by the core looseness. The

underlying reason for this is the modal participation, where the 2nd mode is dominated

by the transformer winding. Since the transformer core also participates in this mode,

the 2nd mode is classified as a coupled mode between the core and winding assemblies.

The orthogonality of vibration modes determines that causes of failure in the

transformer core will not have much effect on the winding-dominated mode, i.e., the 2nd

mode in this study.

The above experimental results verify that core looseness is able to change more

than one mode in the low-frequency range. The dependency of natural frequency shifts

on the causes of structural damage can potentially be employed for damage location.

http://dict.youdao.com/search?q=orthogonality&keyfrom=E2Ctranslation

103

In addition to the frequency shift analysed above, the deviations of both

mechanically and electrically excited FRFs at 100 Hz due to core clamping looseness

are listed in Table 5.1, where a maximum 2.56 dB increase under electrical excitation is

observed. As another “side-effect” of stiffness reduction, the increase in the amplitude

of the FRF is expected. Attention should be paid to the frequency shifts, which would

increase the vibration response when a resonance peak is approaching and vice versa.

However, the variation under two excitations exhibits different sensitivities. The

deviation of the electrically excited FRFs ( H ) depends not only on the stiffness

changes but also on the magneto-mechanical coupling during magnetisation of the

transformer core. With clamping looseness, the internal stress of the silicon steel

laminations is reduced as well, which leads to a weaker magneto-mechanical coupling

and thus smaller magnetostriction [49, 85]. This is another factor influencing the

electrically excited FRFs, as core clamping looseness will decrease the magnetostriction.

Combined with the above factors, the overall effect is to increase the electrically excited

FRFs due to core looseness in this study. It is also worth mentioning that the vibration

response under electrical excitation is more sensitive to core looseness, as can be

concluded from Table 5.1.

5.4.2 Vibration response due to winding looseness

Transformer winding looseness is the second cause of mechanical failures in the model

transformer to be studied. Winding looseness is described by (0)2 2 2(1 )M M with

20 1 to represent different clamping statuses. By gradually reducing the clamping

force with the same percentage increment ( 2 5% ), an overall 35% looseness was

introduced to the transformer winding. The mechanically and electrically excited FRFs

were measured using the same test equipment used in Section 5.1. Figure 5.3 shows the

spatially averaged FRFs under different winding clamping forces. A 5-dB offset from

104

the FRF underneath is also introduced to clarify the picture.

Figure 5.3. Spatially averaged FRFs of the transformer vibration due to (a) mechanical and (b) electrical excitations with winding clamping looseness.

As can be seen in Figure 5.3, four natural frequencies can be recognised from

the electrical and mechanical FRFs below 120 Hz. The difference comes from their

variations in the presence of winding clamping looseness. When the mechanical

parameter vector changes from 2 0 to 2 35% by reducing the winding clamping

forces, a gradual decrease occurs at the 2nd natural frequency. Meanwhile, the natural

20 40 60 80 100 120 140-20

-10

0

10

20

30

40

Frequency [Hz]

FRF

[dB

]

(a)

2=35%2=30%2=25%2=20%2=15%2=10%2=5%2=0

20 40 60 80 100 120 140-20

-10

0

10

20

30

Frequency [Hz]

FRF

[dB

]

(b)

2=35%2=30%2=25%2=20%2=15%2=10%

2=5%2=0%

105

frequencies of the other modes appear to be unaffected. Compared with the variation of

FRFs in the presence of core looseness, this phenomenon can be regarded as

characteristic for winding anomalies. Similarly, the quantitative variation of the

vibration FRFs due to winding clamping looseness is summarised in Table 5.2. Since

the most affected mode is the 2nd mode at 53 Hz, the frequency response at 50 Hz is

analysed in addition to the natural frequency shift.

Table 5.2. Quantitative variation of the vibration FRFs due to winding clamping looseness.

1(%) 5 10 15 20 25 30 35

nf (%)

1st mode −1.4 −1.4 −1.4 −1.4 −1.4 −1.4 −1.4

2nd mode −1.9 −3.8 −5.7 −6.6 −9.4 −10.4 −11.32

3rd mode −0.6 −0.6 −0.6 0 0 0 −0.6

4th mode −1.5 −0.9 −0.5 −0.5 −0.9 −1.94 −1.94

MH (dB) 50Hz 0.44 2.43 3.65 3.7 2.39 1.97 2.09

H (dB) 50Hz 1.53 2.19 4.07 4.52 4.75 4.24 3.62

As can be seen in Table 5.2, the appearance of winding clamping looseness

leads to a general decrease of all four natural frequencies. The most affected mode is the

2nd mode at 53 Hz, which has an 11.32% (6 Hz) decrease at maximum looseness. This

experimental result is fully expected since the introduced causes of failure are a natural

reduction in the transformer’s local stiffness. As to why the frequency shift occurs at the

2nd mode, the answer can be found from the modal analysis, where the 2nd mode is

dominated by the winding assembly. The winding clamping looseness mainly causes

stiffness reduction in the winding rather than the core assembly. Since the natural

frequencies are measured in exactly the same way as in the mechanical and electrical

106

excitation cases, the same trends in frequency shift are observed in the electrically

excited cases.

Apart from the analysis of natural frequency shifts, amplitude variations of the

FRFs due to winding looseness are also examined in both the mechanically and

electrically excited cases. The deviations calculated in Table 5.2 are also cumulative

changes relative to the lowest state without looseness ( 2 0 ). Compared to the

vibration FRF of the initial clamping state, a general increase at 50 Hz is found in both

the mechanically and electrically excited cases in the presence of winding looseness. In

particular, the vibration response at 50 Hz firstly increases with the approach of the 2nd

natural frequency and then decreases as it moves far away. The overall increase in the

vibration response can be understood as a result of the reduction of the stiffness in the

winding assembly.

Since the vibration test was performed in a transformer under no-load conditions,

the electromagnetic (EM) force in the winding is caused by the interaction between the

magnetising current and the leakage field. The magnetising current is only few hundred

milliamps and the resulting EM force is relatively small. Excitation caused by EM

forces in the winding is very weak and can be neglected compared to core excitation.

Therefore, the excitation force in these cases remains almost the same, and is mainly

composed of magnetostrictive force in the core. In other words, the changes in

mechanical properties induced by a structural anomaly are responsible for the variations

of the vibration FRFs. Maximum increases of 3.7 dB and 4.75 dB were measured at 50

Hz in the mechanical and electrical FRFs, respectively. Such obvious deviations are

more than enough to be detected in the vibration response–based monitoring methods.

5.4.3 Vibration response due to missing insulation spacers

The third case study is dedicated to one of the causes of insulation faults in the winding

107

assembly. As described in Figure 5.1, insulation damage is simulated by removing a

small portion of the insulation spacers along the longitudinal direction. In this study, the

absence of insulation spacers is catalogued as a mechanical property change since it

indeed alters the mechanical integrity of the transformer structure. The spatially

averaged FRFs of the transformer vibration with missing insulation spacers in the

winding insulation system are presented in Figure 5.4.


and (b) electrical excitations with missing insulation spacers.

20 40 60 80 100 120 140-15

-10

-5

0

5

10

15

20

Frequency [Hz]

FRF

[dB

]

(a)

n=13

n=9

n=5

n=1

n=0

20 40 60 80 100 120 140-20

-15

-10

-5

0

5

10

15

20

Frequency [Hz]

FRF

[dB

]

(b)

n=13

n=9

n=5

n=1

n=0

108

No obvious frequency shifts in this frequency range can be found by visual

examination of the mechanically and electrically excited FRFs, even when all the front

insulation spacers are removed ( 13n ). To quantitatively analyse the variation of the

FRFs due missing insulation spacers, a detailed summary of these frequency shifts is

listed in Table 5.3. Since the missing insulation spacers were introduced in the winding

assembly and the 2nd mode at 53 Hz is dominated by this component, the vibration

response at 50 Hz is also presented in Table 3. The maximum frequency shift in this

case is 2.5 Hz (4.71%) at the 2nd mode, while the other modes appear unaffected. The

underlying reason is also attributed to stiffness reduction in the winding.

Table 5.3. Quantitative variation of the transformer vibration FRFs due to missing insulation spacers.

n 1 5 9 13

nf (%)

1st mode 0 0 0 −1.43

2nd mode −1.9 −2.83 −3.77 −4.71

3rd mode 0 0 0 −0.65

4th mode 0 0 −0.49 0

MH (dB) 50f Hz 0.76 1.97 2.0 2.29

H (dB) 50f Hz 2.51 2.99 3.04 3.47

The analysis of extracted data verified that, although the frequency shifts are

small, the amplitude increases at 50 Hz are pronounced; they are 2.29 dB and 3.47 dB in

the mechanical and electrical FRFs, respectively. These results imply that the amplitude

of the frequency response at certain frequencies can be altered dramatically even with a

small frequency shift when there is a resonance frequency nearby. Similar to the

winding looseness case, these variations in the FRFs are merely caused by changes in

mechanical property rather than excitation differences.

109

Briefly summarising the above observations, it was found that a structural

anomaly in the core could produce considerable variations in the low-frequency range.

Although the amplitude of the FRFs changed dramatically as a result of winding

anomalies, i.e., missing insulation spacers, the sensitivity of the natural frequency shift

is not high. As can be seen in Figure 5.2(a), although the coupled mode at 53 Hz is

dominated by the transformer winding, it is actually the rigid-body movement around

the core bottom yoke. The clamping looseness and missing insulation spacers affect the

connection boundaries in this coupled mode and thus give rise to the above variations in

the FRFs. However, the most affected modal response is anticipated to be at the

winding-controlled modes in the higher frequency range. According to the previous

modal analysis on the same transformer in Chapter 3, the winding-controlled modes are

in a frequency range of >200 Hz. To verify this speculation, the variation of the FRFs

between 120 Hz and 1000 Hz are investigated in the following section.

5.4.4 Variation of the high-frequency vibration response

As was seen in the above discussion, the natural frequencies obtained under the

electrical excitation were the same as under the mechanical excitation. To study the

frequency shift due to winding anomalies, the FRFs of the mechanical excitation are

selected for analysis. The electrical excitation case is not examined owing to the lack of

a high-voltage source with variable frequencies.

(1) Variation of high-frequency response to winding looseness

To study the vibration changes in the higher frequency range, the radial and axial

vibration modes are firstly examined to classify whether they are winding-controlled

modes. The spatially averaged FRFs due to the different winding clamping forces are

presented in Figure 5.5, where the mode shapes for specific resonances are shown as

well.

110

Figure 5.5. Spatially averaged FRFs of the transformer vibration due to winding

looseness in the (a) radial and (b) axial directions.

The winding-controlled modes at around 200 Hz, 400 Hz, 500 Hz, and 600 Hz are all

dominated by the winding assembly, where the participation of the core can be

neglected. As can be seen from Figure 5.5, the natural frequencies at these four modes

all decrease with the development of winding clamping looseness. This observation is

the same as in the low-frequency range. However, the sensitivity of the frequency shift

to the winding looseness is much higher than that in the low-frequency range. Table 5.4

specifies the corresponding frequency shifts of the relevant modes, where a maximum

200 400 600 800 1000 1200

-20

-10

0

10

20

30

40

Frequency [Hz]

FRF

[dB

]

(a)

2=35%2=30%2=25%2=20%2=15%2=10%2=5%2=0

200 400 600 800 1000 1200

-20

-10

0

10

20

30

40

Frequency [Hz]

FRF

[dB

]

(b)

2=35%2=30%2=25%2=20%2=15%2=10%2=5%2=0

(b)

111

32.6% (137 Hz) decrease in the natural frequency can be found for the 2nd winding-

controlled mode in the axial direction. The frequency shifts for the modes with large

radial components are also remarkable, which reach a 20.8% reduction at 35% winding

looseness.

Table 5.4. Natural frequency shifts ( nf ) of the winding-controlled modes due to looseness of the winding clamping force.

2 (%) 5 10 15 20 25 30 35

Radial modes

1st (Hz) −12 −28 −35 −40 −53 −63 −75

2nd (Hz) −13 −28 −42 −50 −58 −70 −86

1st (%) −2.8 −6.8 −8.9 10.3 −13.8 −17 −20.8

2nd (%) −1.9 −4.3 −6.6 −8 −9.4 −11.6 −14.5

Axial modes

1st (Hz) −12 −21 −26 −29 −31 −33 −35

2nd (Hz) −11 −26 −41 −59 −91 −114 −137

1st (%) −5.1 −9.4 −12.2 −13.9 −15 −16.1 −17.3

2nd (%) −2.1 −4.9 −8.1 −11.9 −19.2 −25.7 −32.6

(2) Variation of high-frequency response to missing insulation spacers

As reported in Section 5.3, the frequency shift due to missing insulation spacers is not

obvious since there is only one coupled mode in the analysed low-frequency range and

it is not sensitive to the change in clamping force. Here, the investigation is extended to

a higher frequency range from 120 Hz to 1000 Hz, which covers four winding-

controlled modes, as illustrated in Figure 5.5. Given the high sensitivity of the

frequency shift to looseness of the winding clamping force, finer test steps are adopted

in this case study. Seven missing insulation statuses are equally spaced from 1n to

13n with the same amount of total missing spacers, as conducted in Section 5.3.

Figure 5.6 shows the spatially averaged FRFs between 120 Hz and 1000 Hz of

transformer vibration for different amounts of missing insulation spacers. The shift of

resonance peaks can be clearly discerned at the winding-controlled mode in the radial

112

direction. Detailed percentage variations and absolute frequency shifts in hertz are

summarised in Table 5.5. A maximum 14.2% (45 Hz) decrease is measured at 13n in

the 1st winding-controlled mode in the radial direction. However, the frequency shift

does not appear to be obvious in the axial direction. For the axial mode at 400 Hz, the

frequency shifts at the last three statuses are not listed owing to the local resonance after

7n . This indicates that reducing the clamping force will not only cause the shifts of

natural frequencies, but also allow the observation of extra resonances in the FRF. This

may be another interesting vibration feature that might be useful for transformer

condition monitoring. Either frequency shifts or the appearance of extra resonance

peaks can be related to the looseness of clamping force.

Table 5.5. Natural frequency shifts ( nf ) of the winding-controlled modes due to missing insulation spacers.

n 1 3 5 7 9 11 13

Radial modes

1st (Hz) −3 −4 −9 −17 −26 −39 −45

2nd (Hz) −9 −17 −23 −32 −38 −66 −67

1st (%) −0.8 −1.1 −2.3 −4.9 −7.8 −12.1 −14.2

2nd (%) −1.5 −2.9 −4.0 −5.7 −6.8 −12.3 −12.8

Axial modes

1st (Hz) 0 1 0 −3 −4 −8 −8

2nd (Hz) 2 3 −3 −16 - - -

1st (%) 0 0.5 0 −0.17 −0.23 −4.6 −4.6

2nd (%) 0.5 0.75 −0.75 −4.0 - - -

The detailed percentage variation and absolute frequency shift in hertz are

summarised in Table 5.5, where maximum frequency shifts of 14.2% (45 Hz) and 9.0%

(36 Hz) are measured in the radial and axial directions, respectively. Thus, it has been

verified that the high-frequency modes are more sensitive to the causes of structural

failures. Although the natural frequencies of power transformers vary with different

113

designs and capacities, one can always use a sensitive frequency range for the shift of

natural frequencies in order to monitor the health status of a transformer. Once the

target frequency range is captured, tracing its trend of variation would be the logical

approach the purposes of condition monitoring.

Figure 5.6. Spatially averaged FRFs of the transformer vibration due to missing

insulation spacers in the (a) radial and (b) axial directions.

5.5 Conclusion

In this chapter, a single-phase 10-kVA model transformer was studied as an example to

200 400 600 800 1000 1200-20

-10

0

10

20

30

40

Frequency [Hz]

FRF

[dB

]n=13n=11n=9

n=5n=7

n=3n=1n=0

(a)

200 400 600 800 1000 1200-20

-10

0

10

20

30

40

Frequency [Hz]

FRF

[dB

]

(b)

n=13n=11n=9n=7n=5n=3n=1n=0

114

demonstrate the changes in its vibration response to several winding faults, namely

looseness of clamping forces in the winding and in the core, and the absence of

insulation spacers. The variations of the FRFs due to mechanical parameter changes

were measured using the sweep-sine test and the impact test. For the three different

causes of transformer faults, their influences on the vibration response were examined

by comparing the structural frequency responses of the intact and “damaged”

transformers. The underlying reasons for these variations were analysed.

As expected, the faults were all capable of altering the mechanically and

electrically excited FRFs. To be specific, the occurrence of winding and core looseness,

as well as the absence of insulation spacers decreases the relevant natural frequencies as

a result of the loss of structural stiffness caused by the faults. The maximum 11.32%

decrease in the low-frequency range and 32.6% in the high-frequency range were

measured in the presence of these causes of failure.

With respect to the vibration response at a few forced vibration frequencies, i.e.,

100 Hz, a general increase was observed in these case studies. Possible reasons were

provided in terms of changes in the excitation forces, shifts of neighbouring natural

frequencies, and the loss of structural stiffness.

Compared to the low-frequency FRFs, higher sensitivities to variation were

found in the high-frequency FRFs. Moreover, local resonances would be produced in

the high-frequency range with the development of the causes of failure, as shown in the

missing insulation spacer case.

It is worth emphasising that the dependency of the frequency shift on the causes

of structural failure might be of practical importance in damage detection and

localisation in transformers. The above conclusions show it is possible to utilise the

115

vibration modal parameters for transformer condition monitoring. A study into online

estimation of a transformer’s dynamic property will be discussed in Chapter 6.

116

Chapter 6 Applications of Operational Modal Analysis to

Transformer Condition Monitoring

Previous chapters have shown that a transformer’s modal characteristics can be

identified using the electrically excited frequency response function (FRF) and are

related to structural damage. These results may be useful for diagnosing the causes of

failure modes in power transformers during normal operation. However, they still fall

short in practical applications according to a review of the literature. Successful

identification of the modal parameters for in-service transformers is an important step

for the application of the parameter identification technique to transformer condition

monitoring. Operational modal analysis (OMA) appears to be a suitable tool for this

purpose. In this chapter, the feasibility of OMA in identifying transformer modal

parameters is discussed with special focus on the mechanisms of natural excitation.

OMA is also applied to a 10-kVA transformer with and without artificially designed

structural damage. Good correlation is obtained between the results of OMA and those

of experimental modal analysis.

6.1 Introduction

Power transformer failures can be broadly categorised as having electrical, mechanical,

and thermal causes [43]. The detection of those causes has been attempted by various

methods, of which the vibration-based method is relatively new to this field. Many

works have indicated that vibration monitoring can provide an alternative approach to

assess the mechanical integrity of a transformer [44–47]. Berler et al. [44] introduced a

117

monitoring method based on the observation that the magnitude of transformer vibration

at higher harmonic frequencies increases with core and winding clamping looseness.

However, their method lacks theoretical support and experimental observations were

made in only a limited number of transformers.

In the work by Bartoletti et al. [45], a number of parameters were used to

classify the different states of a transformer. The weighted total harmonic distortion

(WTHD) and the amplitude ratio between the 50 Hz and 100 Hz components ( 50/100R )

were employed to identify aging and anomalies in a transformer. They claimed that a

high WTHD value corresponded to “old” transformers and that a high 50/100R value was

associated with certain transformer anomalies. However, in practice, the same

percentage change in a selected parameter may result from entirely different faults when

a transformer has a number of possible faults.

In a recent publication [46], the relative coefficient, normh , which represents the

distribution of the vibration spectrum, was used as a diagnosis estimator. The

experiment showed that high values of normh within a wide frequency range might result

from degradation of solid insulation and deformation of windings [47]. The criterion

adopted in this method was based on the experimental observation that changes in the

transformer structure from that of a healthy transformer led to an excessive vibration.

The relative coefficient normh was used to describe this excessive vibration. However,

this method depends on a database of faults and needs expert knowledge to interpret the

results. In addition to the aforementioned methods, monitoring based on correlation

analysis was recorded in Ref. [47] and the author claimed that tendency analysis could

be used as well.

It is obvious that the efforts made to analyse the vibration response typical aim

to extract the fault-dependent parameters, which are able to identify the fault origin and

118

its severity. In most cases, an assessment algorithm for a specific transformer needs

plenty of training data in order to form a judgement standard, which may not be

applicable to other transformers

However, owing to the complexity of transformer design and vibration

mechanisms, not to mention general application to in-service units in the power grid, the

extracted parameters often either vary with different transformers or are insensitive to

structural damage,. Fortunately, of the many possible parameters, the modal parameters

of a transformer’s mechanical structure appear to be common to all types of

transformers, and have clear physical significance. As already presented in Chapter 4,

transformer modal parameters can be identified by internal electrical excitations and

they are closely related to several mechanical causes of transformer failure modes. The

occurrence of structural damage affects certain parameters of the dynamic system and

thus affects the modal parameters.

To extract the modal parameters of on-site transformers, the challenge is to find

an appropriate and convenient excitation to excite the transformer’s structural modes, as

most transformers are only excited at discrete frequencies that do not coincide with the

natural frequencies of the modes. Experimental modal analysis (EMA) is the main

approach to obtain transformer modal parameters, as introduced in Chapters 3 and 4. An

external excitation in terms of a mechanical impact or sweep-sine excitation is

necessary to provide force input during EMA. However, owing to safety considerations

of the power grid, an in-service transformer cannot be allowed to undergo such external

excitations, even when it is off-line. In this context, operational modal analysis (OMA),

which can be performed without interrupting the transformer’s normal operation, may

find an application.

119

Although OMA has achieved great success in civil engineering, its application

to transformer condition monitoring has not yet been reported. However, the power

transformer structure shares some similarities with large civil structures, i.e., wind

turbines, large span bridges, and aircraft structures [86–88]. Firstly, they are difficult to

excite by mechanical impact owing to their large dimensions and heavy mass. This

directly leads to the failure of traditional EMA for such large structures. Secondly, they

both undergo random ambient excitations. For civil structures, ambient excitations are

caused by ground microtremors, traffic, and wind loading. Power transformers suffer

from voltage and current variations, which can be another source of ambient excitation

[89].

However, despite the aforementioned common features, there are a few

behaviours specific to the power transformer case. Unlike civil structures, a power

transformer is self-excited by electromagnetic (EM) and magnetostrictive (MS) forces

formed in the winding and core assemblies. It has been verified in Chapter 4 that these

internal forces are able to excite the transformer’s structural resonances. The

disturbance of the natural excitations introduced by energising, de-energising, and

voltage variations gives rise to more information that might be beneficial to OMA.

Inspired by these similarities and particular features, it is proposed to introduce

OMA to power transformers. However, its feasibility should be analysed in advance. In

this chapter, the feasibility of performing OMA on in-service units is analysed in

different operating conditions, including energising, steady-state, and de-energising.

Based on the feasibility analysis, a time-domain Natural Excitation Technique/Ibrahim

Time Domain (NExt/ITD) algorithm, which is verified by a numerical case study, is

employed to perform OMA on a 10-kVA transformer. OMA is further applied on a 10-

120

kVA transformer with artificially designed structural damage. As will be seen, good

correlation is obtained between the estimated outcomes and the EMA results.

6.2 Theoretical background

The basic purpose of transformer OMA is to identify its modal parameters under its

operating conditions. In this study, a combined NExt/ITD method is adopted for

transformer OMA to achieve this purpose. An OMA-based transformer condition

monitoring strategy is developed and presented in Figure 6.1, where the algorithm

processes the vibration response, identifies its modal parameters, and realises the

condition monitoring.

Figure 6.1. Schematics of the OMA-based transformer condition monitoring technique.

The OMA-based condition monitoring begins with the sampling of the vibration

signal. Then, the signal is detrended to remove the DC offset and tendency items. After

that, a judgment step is adopted to determine whether a NExt technique is needed. If the

Start

Vibration Sampling

Detrend

Free Vibration

Comb Filter

Deviation?

NExt

ITD

Warning

Y

N

121

recorded signal is not from the free vibration, then the cross-correlation functions (CCF)

will be calculated by the NExt technique. Finally, the ITD identification is performed

using the free vibration or the CCF obtained from the NExt calculation. It should be

mentioned that the vibration harmonics, whether coming from the nonlinear MS or

electrical noise, should be removed by a comb filter in the forced vibration case. The

filtered vibration response is mainly caused by ambient excitations. The identified

modal parameters will be estimated while referring to the baseline values that are

benchmarks for its healthy status. According to the diagnosis summary, corresponding

actions, e.g., a beep warning, will be generated to attract the attention of the

maintenance technician.

Taking the forced vibration case as an example, this section briefly introduces

the theoretical background of the employed method. Although external ambient

excitation is always present, forced vibration is mainly caused by the EM and MS forces.

Steady-state vibration and transient vibration due to voltage variations and energising

operations are all within the scope of forced vibration.

In the forced vibration case, the NExt technique is employed to calculate the

CCF to substitute the free vibration response required in the ITD algorithm. The

theoretical foundation of the NExt technique is that the CCF is a sum of decaying

sinusoids of the same form as the impulse response function (IRF) of the system. A

brief review of this deduction is introduced in Ref. [90]. The IRF is written as:

1( ) ( ) ( )

tnr

ik ir kr kr

x t f g t d

, (6.1)

where:

0 , 0( ) 1 exp( )sin( ), 0

rr r r

d dr rd

tg t

t t tm

,

122

2 1/2(1 )r r rd n is the damped modal frequency, r

n is the r th modal frequency, r

is the r th modal damping ratio, rm is the r th modal mass, n is the number of modes,

ir is the i th component of mode shape r , and t is time.

The CCF of two responses ( ikx and jkx ) due to a white-noise input at a

particular input point k can be expressed as:

1( ) exp( )cos( ) exp( )sin( )

nr r r r r r r r

ijk ijk n d ijk n dr

T A t T B t T

, (6.2)

where rijkA and r

ijkB are independent of T and are functions of only the modal

parameters containing the second of the two modal summations:

1 0

sin( )exp( ) sin( )

cos( )

r rnijk dk ir kr js ks r r s s s

n n dr r s sr rs d dijk d

Ad

m mB

,

where t .

An important conclusion that can be drawn is that the correlation between

signals is a superposition of decaying sinusoids containing the same damping and

natural frequencies as the structural mode. This similarity allows OMA to be performed

with only the response data without knowing its force inputs. Provided that the CCF has

already been obtained, time-domain identification techniques, e.g., the ITD technique,

the eigenvalue realisation algorithm, and the least squares complex exponential method

can all be exploited to identify modal parameters. In this work, the ITD technique will

be adopted as an example.

A detailed theoretical deduction of the ITD technique can be found in Ref. [91].

However, only a few of its key steps are reviewed in this section to avoid redundancy.

For a multiple-degrees-of-freedom system, the equation of motion for free vibration can

be written as:

123

[ ] ( ) [ ] ( ) [ ] ( ) 0M x t C x t K x t . (6.3)

Assuming the solution is:

21 2 1

tN NN N

x t e

, (6.4)

where:

1 2, , ,T

Nx t x t x t x t ,

1 2 2, , , n ,

21 2, , , NTtt tte e e e ,

x t is the free vibration vector for the displacement,

is the mode shape matrix,

r is the thr eigenvalue, and

N is the mode order.

The response of the thi point at time kt can be written as:

**

1 1( ) ( )r k r k r k

N Mt t t

i k ir ir irr r

x t e e e

, (6.5)

where ir is the thi component of the thr mode shape vector, *( )i N r ir , *

N r r ,

and 2M N .

To construct an M L response matrix, virtual test points are typically employed by

delaying the same response with time t ; see Eq. (6.6). This new response contains the

same dynamic characteristics of the system:

i jn k i kx t x t j t . (6.6)

After repeating 1M times, the response vector at L time points becomes:

124

1 1 1 2 1

1 1 1 2 1

1 2

, , ,

, , ,

, , ,

L

jn jn jn LM L

M M M L

x t x t x t

X x t x t x t

x t x t x t

. (6.7)

Substituting Eq. (6.5) into Eq. (6.7), and letting (t )ik i kx x , the response matrix can be

expressed as:

1 1 1 2 1

2 1 2 2 2

1 2

11 12 1 11 12 1

21 22 2 21 22 2

1 2 1 2

L

L

M M M L

t t tL M

t t tL M

t t tM M ML M M MM

e e ex x xx x x e e e

x x x e e e

.

Cast into matrix form, this is:

M L M M M LX

. (6.8)

Delaying the response of both the real and virtual test points by t gives:

( )

1 1

r k r k

M Mt t t

i k i k ir irr r

x t x t t e e

, (6.9)

where r tir ire

.

Therefore, the response matrix of the delayed response becomes:

M LM MM LX

. (6.10)

From Eq. (6.9):

M M M MM M

, (6.11)

where M M

is a diagonal matrix with diagonal elements

r tr e .

Combining Eq. (6.8) and (6.10), and eliminating gives:

A , (6.12)

where A is a unilateral least squares solution of A X X .

125

Equation (6.12) is a standard eigenvalue problem in which its eigenvalues are

related to the natural frequencies of the system and the eigenvectors pertain to the mode

shapes. By solving the eigenvalue problem, the dynamic properties of the system can be

inferred.

For the free vibration case, the identification procedure becomes more

straightforward. The vibration response can be directly utilised as input data in the ITD

algorithm. In this case, it is not necessary to calculate the CCF using the NExt technique.

6.3 Feasibility analysis

Unlike EMA, OMA is based only on the output response without measuring the input,

which is assumed to be an excitation with random features or a known mathematical

expression, such as an impulse excitation. As a result, OMA is only suitable for

extracting the modal parameters of power transformers if such an excitation occurs and

it can adequately stimulate the vibration of the transformer structure. In practice,

however, transformer structures are excited by their electromagnetic inputs and loading

variation, and by seismic and geomagnetic excitations. Natural excitations vary

depending on the operating (including energising, steady, and de-energising states) and

environmental conditions. If any of these excitations satisfy the requirements of OMA,

then the modal parameters of the transformer can be extracted for analysis.

126

Figure 6.2. Time-frequency spectra of a 10-kVA transformer in (a) energising, (b)

steady, and (c) de-energising states.

Figure 6.2 presents the time–frequency spectra of a 10-kVA transformer in the

energising, steady, and de-energising states. The same time span (3.5 s) is chosen for

each state. Figure 6.2(a) shows the background transformer vibrations before energising

begins at t = 1 s. The transformer’s modal characteristics are often difficult to excite

during this period, and the performance of OMA would be poor owing to the low

signal-to-noise ratio. Energising the transformer involves a large transient excitation

between 1 and 1.5 s, which contains many frequency components (Figure 6.2(a)). This

broadband vibration response carries useful information about the transformer’s modal

characteristics and the significant magnitude of steady-state vibration at 50 Hz and its

harmonics. Compared to the transient vibration during de-energising, the transient state

during energising has a longer settling time. However, the steady-state vibration

components typically overshadow the broadband vibration response, requiring pre-

processing to remove the steady-state components before applying OMA. Vibration in

the steady state, however, only consists of discrete frequency components at 50 Hz and

their harmonics (Figure 6.2(b)), while the vibration response at other frequencies is at

(a) (b) (c)

127

the background level. Therefore, the transformer’s modal characteristics cannot be

sufficiently excited in the steady state.

Figure 6.3. Vibration waveform of a 10-kVA transformer in the (a) energising, (b)

steady-state, and (c) de-energising conditions.

De-energising gives rise to a short transient vibration, which declines rapidly

after ~0.15 seconds. However, this is a free vibration since the internal electromagnetic

excitation was completely terminated by switching off the electromagnetic circuit of the

transformer. In this case, the ITD algorithm can be applied for OMA of the transformer

modal parameters. The time-domain waveforms of transformer vibration in the three

operating conditions are presented in Figure 6.3, where transient vibration can be clearly

observed in the time domain. The difference in settling time between energising and de-

energising vibration becomes more straightforward in Figure 6.3.

To verify whether the aforementioned vibration characteristics, especially the

free vibration captured from the transformer’s de-energising operation, can be found in

larger transformers, the vibration of a 15-MVA 3-phase station transformer was

analysed in the energising, steady, and de-energising states.

0 1 2 3 4 5-10

0

10

2.9 3 3.1 3.2 3.3-2

0

2

2.9 3 3.1 3.2 3.3-2

0

2Vib

ratio

n [m

/s-2

]

Time [s]

128

Figure 6.4. Time-frequency spectra of a 15-MVA transformer in the (a) energising, (b)

steady, and (c) de-energising states.

The aforementioned vibration characteristics can also be found in larger

transformers. Figure 6.4 shows the time–frequency spectra of the vibration of a 15-

MVA 3-phase power transformer during the energising, steady, and de-energising states.

The vibration response during energising comprises 50 Hz and its harmonics, and

gradually settles to the steady state. Similar to a single-phase distribution transformer,

de-energising the 15-MVA unit generated free vibration in the transformer structure

within a similar time scale of 0.15 s. Although the underlying excitation mechanisms of

transformer vibration are the same for both, it should be noted that the background

vibration of the on-site transformer is higher than that of one in a laboratory

environment. From a comparison of the waveforms of transformer vibration in different

statuses (Figure 6.3 and Figure 6.5), it is clear that the vibration response is larger for a

15-MVA power transformer, even though they share the same steady-state and transient

vibration features.

(a) (b) (c)

129

Figure 6.5. Vibration waveform of a 15-MVA transformer in the (a) energising, (b)

steady-state, and (c) de-energising conditions.

In conclusion to the above analysis, transformer vibration, at least the de-

energising vibration, can be adopted for OMA as a potential condition monitoring

method. Whether forced vibration can be adopted for OMA will be further discussed in

Section 6.4.

6.4 Operation verification

6.4.1 OMA for a 10-kVA transformer

Based on the feasibility analysis in Section 6.3, the ITD algorithm is used to extract the

natural frequencies of a 10-kVA (415/240 V) distribution transformer from its vibration

during the de-energising process. With regard to the forced vibration case, OMA is also

conducted, assuming that the modal characteristics can be excited through ambient

excitation or voltage and current variations. Through this trial identification, it is hoped

that the feasibility of OMA in a forced vibration scenario can be practically validated.

0 2 4 6 8 10-20

0

20

0 0.1 0.2 0.3 0.4-5

0

5

Vib

ratio

n [m

/s-2

]

0 0.1 0.2 0.3 0.4-2

0

2

Time [s]

130

(1) Modal identification in the free-vibration condition

In this case, the data input for the OMA can be measured during the de-energising

process. Therefore, it is still within the scope of OMA and is meaningful for transformer

condition monitoring. The results identified through OMA are presented in Figure 6.6 in

a standard stabilisation diagram, where the vibrational power spectral density (PSD) is

also included for comparison. The symbol “ ” represents the OMA-identified natural

frequencies of the transformer. To avoid ambiguities caused by the ill-conditioned

eigenmatrix, various numbers of modes are used for identification. As a result, a trace of

the natural frequencies can be mapped out and final values are determined via averaging.

A practically determined range of modal damping ratios from 0.01 to 0.3 is selected as a

constraint for the identification.


transformer. The solid line is the spatially averaged PSD.

As can be seen in the PSD plot, the peak-picking method is able to discern up to

seven modes in the analysed frequency range, though only four modes were actually

reported in Chapter 4. These four natural frequencies are 35 Hz, 53 Hz, 77 Hz, and 103

131

Hz obtained from the EMA experiment. It is apparent that the peak-picking method falls

short in accurately discerning the natural frequencies. However, the employed method

appears to be much better for this purpose. As can be seen from the comparison in Table

6.1, the natural frequencies identified by the employed method are closer to the real

values with a maximum deviation of 2.73% at the 3rd mode.

Table 6.1. Comparison of natural frequencies (in hertz) from EMA and free-vibration-based OMA.

Mode EMA Free Vibration Error (%)

1 35 35.2 0.57

2 53 53.6 1.13

3 77 79.1 2.73

4 103 101.4 1.55

(2) Modal identification in the forced-vibration condition

Figure 6.7. The steady-state vibration, filtered response, and calculated correlation

function of a 10-kVA transformer.

In the forced vibration case, the data input for OMA can be measured in the

transformer’s on-load status. Before performing modal identification, the vibration

0 0.1 0.2 0.3 0.4 0.5-0.5

0

0.5

Acc

.1 [m

/s-2

]

0 0.1 0.2 0.3 0.4 0.5-0.2

00.2

Acc

.1 [m

/s-2

]

0 0.1 0.2 0.3 0.4 0.5-1

0

1x 10-3

Time [s]

Original vibration

Filtered vibration

Correlation function

132

component at 50 Hz and its harmonics are removed by a comb filter. The filtered

vibration is then adopted in the NExt technique to calculate the CCF for the ITD

identification. The original vibration response, filtered vibration, and the calculated

correlation function are shown in Figure 6.7. The original vibration reduces

significantly after applying a comb filter up to 1000 Hz. However, it is still larger than

the background vibration, which is merely caused by the ambient excitation. This

indicates that there are also a few harmonic components >1000 Hz involved in the

vibration response. However, this does not affect identification in the low-frequency

range in this study. As mentioned in the theoretical background, the CCF is actually a

series of decaying sine waves. This can be witnessed in the bottom diagram in Figure

6.7. Based on the ITD algorithm, the natural frequencies are identified using the

calculated CCFs. The OMA results are presented in a stabilisation diagram in Figure 6.8.


transformer. The solid line is the spatially averaged PSD.

In this case, the results of the peak-picking method for the PSD curve are

misleading as well. Interference from the ambient environment and contamination of the

133

background noise are the possible reasons for this. In contrast, the NExt/ITD-based

OMA provides more reliable results with a maximum deviation of 8.2% at the 4th mode.

A more detailed error comparison can be found in Table 6.2.

Table 6.2. Comparison of the natural frequencies (in hertz) from EMA and forced-vibration-based OMA.

Mode EMA Forced Vibration Error (%)

1 35 34.2 2.29

2 53 - -

3 77 83.34 8.23

4 103 105.3 2.23

In addition, the performance of free-vibration-based and forced-vibration-based

OMA are compared in Table 6.1 and Table 6.2. Although the natural frequencies can be

identified in both scenarios, the prediction accuracy of the free-vibration-based OMA is

better than that of the forced vibration case. To improve the identification accuracy in

both cases, a long period of sampling is recommended to ensure that all modes of

interest are sufficiently excited. Since condition monitoring is performed continuously,

the natural frequency calculated from neighbouring measurements can be compared to

confirm the identified result as well.

6.4.2 Structural damage detection based on transformer OMA

In order to demonstrate the ability of OMA to locate structural damage and estimate the

severity in a transformer, OMA is applied to the same 10-kVA transformer with

artificially designed structural damage in this section. The artificially designed damage

is introduced in the form of clamping looseness in the core and winding assemblies.

These are realised by adjusting the clamping bolts in the corresponding positions,

134

shown in Figure 3.10. Using the aforementioned OMA strategy, the modal parameters

are identified in each case study. Since the free-vibration-based OMA exhibits better

accuracy, the following OMAs are conducted with the de-energising vibration.

(1) Modal identification in the presence of core looseness

In the first case study, a 50% clamping-force looseness is introduced to the left and right

limbs, respectively. The results identified by OMA are presented in Figure 6.9 in a

standard stabilisation diagram, where the vibrational PSD of the healthy transformer is

used to indicate its original resonance peaks.


transformer with core looseness. The solid line is the spatially averaged PSD without clamping looseness.

The symbols “” and “+” represent the OMA-identified natural frequencies of

the transformer with left- and right-limb looseness, respectively. The same range of

modal damping ratios from 0.01 to 0.3 is selected as a constraint for the identification,

as with the healthy status. For the left-limb looseness, the natural frequency of the 3rd

mode reduces from 79.1 Hz to 72.7 Hz, resulting in an 8.1% relative reduction (as

135

shown by 1f in Figure 6.9). A 6.5% reduction in the natural frequency of the 4th mode

is observed (as shown by 2f in Figure 6.9), due to the same percentage of clamping

looseness in the right limb.

(2) Modal identification in the presence of winding looseness

The second case study introduced 25% and 50% winding looseness to the transformer.

This case study aims to validate the ability of the employed method to estimate the

damage severity. In Figure 6.10, “” and “+” represent the natural frequencies with 25%

and 50% looseness, yielding 6.53% and 11.2% reductions in the natural frequency of

the 2nd mode, respectively.


transformer with winding looseness. The solid line is the spatially averaged PSD without clamping looseness.

6.5 Conclusion

Given the successful application of OMA methods to condition monitoring in civil

structures, this chapter used OMA to extract the natural frequencies of a transformer.

Vibration analysis of a 10-kVA distribution transformer and a 15-MVA power

136

transformer indicated that the vibration associated with the de-energising state is

suitable for OMA. An OMA-based monitoring strategy was developed based on the

NExt/ITD method, which also involved special pre-processing of the vibration response.

Two case studies on a 10-kVA transformer with an ITD-based OMA showed good

agreement with EMA results. Limited to the employed method, the free-vibration-based

OMA exhibited better identification accuracy than the forced-vibration case, which

implied that the OMA performance can be improved significantly while considering the

standard operating events of in-service transformers. In addition, the ability of OMA to

locate structural damage and estimate damage severity in a transformer was

demonstrated in another two case studies. It appears that the proposed OMA-based

monitoring strategy may be an effective technique for transformer condition monitoring.

137

Chapter 7 Conclusions and Future Work

7.1 Conclusions

This thesis has been devoted to studies of transformer vibration and its applications to

transformer condition monitoring. The vibration responses of a small-distribution

transformer were studied both numerically and experimentally. The accurate modelling

of the electromagnetic (EM) force and vibration response of the transformer core and

winding was one of the main focusses of this thesis. Variations in vibration and their

sensitivity to different structural faults were studied experimentally under both

mechanical and electrical excitations.

Chapter 2 demonstrated that the 2D modelling of EM forces in practical

transformers was inaccurate. A 3D model is necessary to accommodate the asymmetric

boundary of the magnetic field and improve the modelling accuracy. This is because the

asymmetrically located transformer core, magnetic shunts, and metal tank all affect the

leakage field and consequently the resulting EM forces. Their effects on the EM forces

were numerically investigated and found to be significant. It was shown that the EM

forces of the transformer are dependent on the shunt shape, position, and dimension.

The presence of magnetic shunts, irrespective of type, increased the radial EM force on

one hand and decreased the axial EM force on the other hand. The areas where the EM

forces were most affected were middle and ends of the winding assembly, where the

maximum EM forces occurred. This observation is useful to practicing engineers in the

transformer industry in designing winding strength and magnetic shunts.

138

A numerical modelling strategy based on the FE method was used in Chapter 3

to predict the vibration response of core-form transformers consisting of winding and

core assemblies. Experimental modal analysis (EMA) was used to verify the numerical

model, as well as the necessary simplifications during creation of the model. The results

showed that the model possessed reasonable accuracy for studying the general features

of transformer vibration. Moreover, they showed that this modelling approach would be

applicable to a broad category of transformers owing to the portability of the FE method.

With the modelling strategy in place, the vibration frequency response of a 10-

kVA small-distribution transformer was calculated. Good agreement with the

experimental results was found. Based on the 3D FE model, three types of structural

anomalies in transformer winding were introduced into the FE model of a transformer: 1)

local deformation, 2) winding tilting, and 3) winding twisting.

During the EMA model verification, the modal characteristics of a core-form

power transformer were discussed thoroughly. The transformer vibration modes were

classified as winding-controlled, core-controlled, and winding/core-coupled modes.

This approach made the description of transformer vibration more specific. The

transformer resonance modes were not always in the high-frequency range despite the

fact that transformers are mostly constructed from copper and steel materials with high

stiffness. Experimental observations and numerical simulations both showed that the

low-frequency modes were usually related to the core-controlled and winding/core-

coupled modes. A straightforward observation is that the transformer vibration cannot

be treated as a lumped parameter system as far as its dynamic responses are concerned.

Instead, the transformer winding behaves more like a cylindrical structure with both

ends constrained. As an alternative method to factory testing, FE analysis provides an

139

alternative method to estimate the dynamic response of a transformer under simulated

structural damage.

Chapter 4 presented the first experimental comparison between the mechanically

and electrically excited frequency response functions (FRF) of a small-distribution

transformer. The comparison demonstrated that both mechanically and electrically

excited FRFs carry information about the modal characteristics of the transformer

structure. This result not only has academic value, but also has practical significance

because the modal characteristics of a transformer structure are related to the

transformer’s mechanical faults. Experimental evidence was also provided to show that

the faults (such as a reduction in core clamping force) caused the changes in the

resonances of the measured FRFs. The difference between mechanically and electrically

excited FRFs was also explained. In particular, the effects of hysteresis of the core

material on the electrically excited FRFs under different levels of voltage excitation

were discussed.

Although the above results were obtained from studying a small distribution

transformer, the methods of extracting the modal characteristics from mechanically and

electrically excited FRFs and identifying the mechanical faults using the measured

modal characteristics can be applied to all types of transformers because the vibration

properties of transformers of different sizes are all controlled by the same mechanical

principles. The resonance phenomena of the transverse modes in small-distribution

transformers and the explanation of these enhance the traditional understanding of the

frequency range of resonances in transformer core vibration.

Although practical transformers vary widely in their mechanical and electrical

structures and operating details (e.g., loading characteristics and location of the voltage

tap-changer), the key features of the FRFs and the differences between them under

140

different excitations still apply. They provide a useful understanding of the modal

response of a transformer when vibration-based condition monitoring techniques are

used for mechanical fault detection in a transformer.

Along the direction of research started in Chapter 4, the changes in the vibration

FRFs due to changes in mechanical parameter were measured through a sweep-sine test

and an impact test in Chapter 5. For three different causes of transformer faults, i.e.,

winding looseness, core looseness, and missing insulation spacers, their effects on the

vibration response were examined by comparing the structural frequency responses of

the intact and “damaged” cases. The underlying reasons for these variations were

analysed related to each case study.

As expected, the introduced faults were all capable of causing changes in the

mechanically and electrically excited FRFs. To be specific, the occurrence of winding

and core looseness, as well as the absence of insulation spacers decreased the natural

frequencies of the transformer structure, owing to the loss of structural stiffness caused

by the faults. A maximum decrease of 11.32% in the low-frequency range and 32.6% in

the high-frequency range were measured in the presence of these faults. With respect to

the vibration response at a few forced vibration frequencies, i.e., 100 Hz, a general

increase was observed in these case studies. Possible reasons were given in terms of

changes in neighbouring natural frequency shifts and loss of structural stiffness.

Compared to the low-frequency FRFs, higher sensitivities of the shift in natural

frequencies to the faults were found in the high-frequency FRFs. Moreover, local

resonances would be produced in the high-frequency range with the development of

faults, as shown in the case of missing insulation spacers. It is worth emphasising that

the dependency of the frequency shift on the transformer structural faults might be of

practical importance in damage detection and localisation. The above results imply that

141

it is possible to utilise the vibration modal parameters for transformer condition

monitoring.

Chapter 6 demonstrated the successful application of operational modal analysis

(OMA) methods in extracting the natural frequencies of a transformer without

measurement of any electrical input. Vibration analysis of a 10-kVA distribution

transformer and a 15-MVA power transformer indicated that the vibration associated

with the de-energising state is suitable for OMA. An OMA-based monitoring strategy

was developed using the Natural Excitation Technique/Ibrahim Time Domain

(NExt/ITD) method, which also involves special pre-processing of the vibration

response. Two case studies on a 10-kVA transformer with an ITD-based OMA showed

good agreement with EMA results. The free-vibration-based OMA showed better

identification accuracy than the forced-vibration case, which implies that the OMA

performance can be improved significantly while considering the standard operating

events of in-service transformers. In addition, the ability of the OMA to locate structural

damage and estimate severity of damage in a transformer was demonstrated in another

two case studies. Thus, it appears that the proposed OMA-based monitoring strategy

may be an effective technique for transformer condition monitoring.

In conclusion, this thesis supposes a substantial contribution to the knowledge

on the field of transformer vibration and transformer health monitoring. Its original

contribution and achievement can be summarized as follows:

1. Implementation of the theoretical comparison of transformer leakage field

between the DFS and FE methods for a single-phase power transformer.

This work verifies the prevailing FE method, which was employed to

investigate the influence of transformer tank and magnetic shunts with

different geometries on transformer winding EM forces. The simulation

142

results and corresponding technical conclusions would be useful for

transformer winding and magnetic shunt design.

2. Correlation of results of mathematical modelling, using the finite element

method, with results obtained during the vibroacoustic measurements on the

actual model of a single-phase 10-kVA power transformer. This work

demonstrates the possibility of FE method in transformer vibration

modelling and failure modes analysis. The proposed strategy can be

extended to transformer mechanical fault simulation, which benefits the

establishment of vibration database for various mechanical faults.

3. Investigation on the FRFs of a 10-kVA power transformer and their changes

induced by transformer mechanical failure causes. Spare effort has been

made to estimate transformer’s vibration characteristics under electrical

excitaion, which is the actual form in its operating conditions. These

experimental results and pointed conclusions enlightened the onsite

application of transformer modal parameter identification.

4. Proposition of an innovative diagnosis method, which enables estimation of

the technical condition of the transformer active part, based on the

measurement, analysis and mathematical modelling of transformer vibration.

The proposed OMA method, which can be utilized under transformer

operating conditions, holds a great potential in industrial application for

transformer condition monitoring.

7.2 Future prospects

This study conducted extensive research on transformer vibration and its application to

condition monitoring. Although it bridged several gaps in this area, many aspects still

have not yet been considered and discussed.

143

With respect to the modelling of EM forces in a transformer, the saturation and

hysteresis characteristics of the transformer core have not been discussed. The accuracy

of the models of these forces can be further improved if these characteristics are

properly included in the calculation. Modelling magnetostriction in the transformer core

and EM forces between laminators is another challenging question for future study.

The effect of fluid on transformer vibration is definitely an interesting topic of

research, which was not been included in this thesis owing to the limited time allowed

for this PhD research. Experimental and numerical studies in this area would be of great

importance to the understanding of the flow of transformer vibration and the

distribution of tank vibration. Modelling of transformer vibration considering the effect

of insulation fluid will be the next milestone on this path of research. In addition to

transformer vibration studies, the noise emission problem will be scheduled for future

research. After the modelling of the vibration of a power transformer, the modelling of

its external sound field is of significant importance to transformer manufacturers.

For vibration-based transformer condition monitoring, more effort will be made

towards the accurate identification of a transformer’s modal parameters by optimising

the OMA methods. It is hoped that the application of this method can be realised in a

compact manner for practical power transformers.

All the above works are expected to be extended to large power transformers

and thus deepen our understanding of transformer vibration and noise problems.

144

Appendix A Further Discussion of Transformer Resonances and Vibration

at Harmonic Frequencies

A.1 Transformer Resonance

Power transformers are constructed of heavy metal materials. The natural frequencies of

a structure made of heavy metal materials are commonly estimated at a few hundred

hertz. However, low-frequency resonances was observed in a 10-kVA small distribution

transformer in Chapter 4. Excluding the rigid body movement relative to its supporting

boundaries, a 35-Hz bending mode was found to be a fundamental resonance of a three-

phase three-limb core-type transformer. This result is actually a challenge to the

traditional understanding. To investigate the causes of this low-frequency resonance, a

series of modelling simulations based on the finite element (FE) method were

conducted. The calculated mode shapes at the first resonance frequency always agree

well with the experimental results. However, the corresponding natural frequency is

much higher than measured.

Given the lamination of the transformer’s core assembly, it is suspected here that

the material properties in each direction might be different as it is clamped by a pair of

metal brackets in one direction. To verify this conjecture, a specimen test based on

vibration modal analysis was performed with the aim of obtaining the anisotropic

material properties of laminated assemblies.

Figure A1 presents the in-plane and off-plane frequency response functions

(FRF) of the test specimen. From the FRFs measured in the in-plane direction, the

fundamental resonance frequency can be easily discerned at 2654 Hz in the tightest

condition. However, in the off-plane direction, it is difficult to identify the natural

145

frequencies owing to the high modal damping, not to mention the exact value of the

fundamental natural frequency.

Figure A1. Vibration FRFs of the test specimen measured in the (a) in-plane and (b) off-plane directions with different clamping statuses.

To better identify the natural frequency for the off-plane Young’s modulus

calculation and exclude the possible influence of the hanging boundary, another eight

groups of repeated tests were performed with gradually reduced clamping forces under

the same hanging boundary. As anticipated, the resonance peak in the in-plane direction

drops to 2638 Hz while the 1st natural frequency in the off-plane direction incrementally

decreases from around 130 Hz to 60 Hz. The frequency shifts in both directions verify

that these peaks are due to structural resonance rather than boundary effects.

Based on the fundamental natural frequencies measured in the in-plane and off-

plane directions, the Young’s modulus in each direction can be calculated according to

the Euler–Bernoulli beam theory with its natural frequencies calculated by:

2

40

n nEILA L

, (A1)

where E is the Young’s modulus, I is the area moment of inertia, is the silicon-iron

(SiFe) density, 0A is the cross-sectional area, and L is the beam length. The value of the

constant nL at the fundamental natural frequency is 4.73 in the free–free condition.

146

A significant difference in the calculated Young’s modulus is noticed in the in-

plane direction (158.6 GPa) and in the off-plane direction (1.8 GPa). The in-plane

Young’s modulus is almost equivalent to that of the SiFe material. In contrast, the off-

plane Young’s modulus is two orders lower. This strong anisotropic property of the core

assemblies attracts attention and stimulates the desire to find the underlying reasons for

the low-frequency resonances of the power transformer.

As reported in Chapter 4, the mode shape at the fundamental frequency of a

three-limb core-type transformer is actually the first-order bending of the side limbs.

Given the mode symmetry and support–free boundary conditions, it is possible to

estimate the fundamental natural frequency using cantilever beam theory with the

constant nL equal to 1.875 in Eq. (A1). Special attention should be paid to the

equivalent beam length, which must extract the height of the bottom yoke as well since

this part is fully constrained by jointing with the bottom yoke.

The fundamental natural frequency is calculated to be 31.1 Hz and 316.1 Hz by

utilising the off-plane and in-plane Young’s modulus values. It is apparent that the

natural frequency calculated by the off-plane modulus is closer to the measured value

and that the calculation based on the in-plane modulus is definitely incorrect.

A2. Transformer Vibration at Harmonic Frequencies

For an in-service transformer, the vibration response is mainly at a few discrete

harmonic frequencies. The vibration-based online condition monitoring is mostly based

on these vibration responses. Considering the odd harmonics involved in the power grid,

the vibration distribution at the 1st, 3rd, 5th, and 7th harmonics is selected for analysis.

Figure A2 presents the vibration velocity distribution of a 10-kVA distribution

transformer at 100 Hz, 300 Hz, 500 Hz, and 700 Hz. Taking the vibration at 100 Hz an

example, the velocity distribution is dominated by the 4th mode, which is the bending of

147

the right limb. As can be seen in Figure A2, the effect of winding vibration becomes

more pronounced in the high-frequency range.

100 Hz

300 Hz

500 Hz

700 Hz

Figure A2. Vibration velocity distribution of a 10-kVA transformer at the 1st, 3rd, 5th, and 7th harmonics.

A3. Transformer Vibration at Specific Test Points

The vibration frequency responses (VFR) at T01, T07, T25, T33, and T40 are presented

in Figure A3, where the absolute FRF amplitudes of T33, T25, T07, and T01 are each

scaled up by 20 dB to better illustrate the VFRs. The resonance peaks can be clearly

recognised at each point. However, the VFR to both electrical and mechanical

excitations exhibits an obvious location dependency. Although the vibration responses

at different locations vary from each other, they are all composed of superpositions of

structural modes. A more detailed modal analysis can be found in Chapter 3.

The vibration responses to mechanical excitations are numerically calculated via

the FE simulation. The VFRs at the same points are shown in Figure A4.

148

Figure A3. Vibration frequency responses at specific test points to (a) electrical and (b)

mechanical excitations.

Figure A4. Vibration frequency responses at five points calculated by the FE method.

20 40 60 80 100 120-20

-10

0

10

20

30

40

Frequency [Hz]

FRF

[dB]

T40T33T25T07T01

20 40 60 80 100 120-40

-20

0

20

40

60

80

100

Frequency [Hz]

FRF

[dB]

T40T33T25T07T01

20 40 60 80 100 120-20

0

20

40

60

80

100

Frequency [Hz]

FRF

[dB]

T40T33T25T07T01

149

Appendix B Voltage and Vibration Fluctuations in Power Transformers

B1. Introduction

Transformer vibrations are mainly generated in the transformer core and windings by

electromagnetic and magnetostrictive forces, and there is an enormous body of research

in this area [B1–B5]. Magnetisation in the transformer core area is well known as a

source of magnetostriction and core vibration [B6]. With respect to the magnetostatic

force between individual sheets of the laminated core, the attractive or repulsive forces

are also deemed to be a possible source of core vibration [B7, B8]. It should be noted

that both magnetostriction and magnetostatic forces in the transformer core are closely

related to the primary voltage, which is the induced voltage in the primary winding and

determines the magnetic field in the core. The only force applied to the transformer

winding is the electromagnetic force. It is very straightforward to show that

electromagnetic forces are proportional to the square of the load current. Since the

consumption of electricity in local power grids typically occurs randomly, the

transformer load varies over time, which induces voltage fluctuation. The corresponding

change in the magnitude of the secondary voltage, expressed as a percentage of the

rating voltage, is defined as voltage fluctuation [B9]. Figure B1 shows the voltage

fluctuation (RMS) time-history of a 500-kV/250-MVA power transformer over a period

of one month.

150

Figure B1. Time-history of voltage variation around the rating voltage of a 500-kV/250-MVA power transformer.

A voltage variation of around 3% was found from the measurements. Voltage

fluctuation of a power transformer is permitted within a certain range, and is determined

by fluctuations in the source voltage and loading current. By causing a change in core

vibration, it has a direct influence on the transformer’s dynamic properties. Because

voltage fluctuation in an electrical network will also cause changes in the current in the

winding, winding vibration can also be a cause of such changes.

In light of the previous research, the mechanical integrity of both windings and

core stacks is vitally important to a transformer’s normal operation. Mechanical

looseness and winding deformation have been studied as typical causes of the loss of a

transformer’s mechanical integrity for decades.

As a transformer ages, the mechanical endurance of the paper insulation is

considerably decreased. Therefore, for a given clamping clearance, the clamping

pressure will vary according to the expansion or shrinkage tendency of the cellulose

[B10]. Under extremely harsh conditions, electromagnetic forces can cause deformation

0 5 10 15 20 25 300.5

1

1.5

2

2.5

3

3.5

Time [day]

Vol

tage

Var

iatio

n [%

]

151

damage to the windings. The structural changes caused by thermal, chemical, and

mechanical deformations would all accumulate over time, and eventually result in

disaster if not regularly maintained. Therefore, research into failure detection methods

has become increasingly popular and has precipitated the development of fault

prognosis.

The vibration features associated with the changes in mechanical parameters of a

transformer’s winding and core can be employed as a useful fault detection tool [B11–

B14]. Currently, variations in transformer vibration are mainly explained by changes in

the transformer’s structure. However, when the voltage varies in a transformer circuit,

the excitation forces, which generate transformer vibration, also change accordingly.

The voltage fluctuation arising from the power source or loading shifts will be

considered as an indispensable factor when analysing vibration features. This appendix

will focus on the relationship between vibration variation and voltage fluctuation in

power transformers based on a 10-kVA single-phase transformer.

B2. Experimental Setup

Figure B2. The experimental rig and schematic of the test procedure.

152

The test transformer was a 10-kVA single-phase transformer with rating

voltages of 415/240 V. Its nominal current was 20 A in the primary winding (240 turns)

and 35 A in the secondary winding (140 turns). Winding clamping pressure was

uniformly supplied by four bolts through two resin pressboards. The transformer core

was formed from a stack of 0.27-mm-thick grain-oriented silicon-iron (SiFe) sheets. At

each joint region, overlapping was created using the conventional single-step-lap

method. The core stack was fixed in place by sets of metal brackets clamped with eight

bolts.

Figure B3. Measurement locations.

The transformer assembly is shown in Figure B2. A variable transformer was

employed to supply desired input voltages (0 to 460 V) to the model transformer. At the

low-voltage (LV) end, a group of heaters were connected in parallel to act as resistive

loads. During the tests, the model transformer was fed with variable voltages and

operated with a series of loading combinations. Two kinds of mechanical failure were

introduced to the model transformer in terms of winding and core clamping looseness.

For all of the above cases, both the high and low voltages and the transformer’s

vibrations were recorded for further analysis. The measurement of these voltages was

performed using two step-down transformers to adjust the high voltage (HV) into an

acceptable range for the DAQ card.

153

The vibration of the transformer was measured using eight high-sensitivity

accelerometers (IMI, model 601A12), attached to eight locations on the transformer.

Their locations are shown in Figure B3. The acceleration signals were amplified and

digitalised by a DeltaTron Conditioning Amplifier (Bruel & Kjaer) and a USB 6259

DAQ card (NI). Each limb of the core was wound tightly by a belt coil (seven turns) to

measure the main magnetic flux in the core while excluding the influence of leakage

flux. All of the tests were performed at room temperature.

B3. Results and discussion

B3.1 Voltage fluctuation and its relation to transformer vibration

Voltage fluctuation in the transformer circuit mainly comes from instability in the

power source and changes in transformer loading status. In this section, the voltage

fluctuations induced by those factors will be studied together with transformer vibration.

As depicted in Figure B1, a voltage variation time-history of an in-service power

transformer (500 kV/250 MVA) was recorded in September 2010. The maximum

voltage difference reached 13.4 kV under normal operating conditions without any

customer accidents. During this time, the loading status varied with customer demands.

In the absence of automatic voltage regulating equipment, the transformer voltages

varied with the loading variations. This normal voltage fluctuation commonly appears

in most power grids and normally does not affect industry applications. However, the

issue of whether its effects on transformer vibration can be neglected has not yet been

verified. The following discussion, based on the experimental study of a model

transformer, could provide a useful and instructive understanding of this topic.

B3.1.1 Flux measurement and its engineering implementation

154

Voltage fluctuation in an electrical network can directly affect the magnetising process

in the transformer core, which can be reflected by differences in flux density and

distribution. Consequently, the magnetic and magnetostrictive forces are greatly

affected. Therefore, core vibrations induced by magnetostatic forces and

magnetostrictive effects will be changed at the same time. Since the transformer’s HV

and LV windings both enclose a certain volume of a non-ferromagnetic zone, the

leakage flux within these areas contributes to the induced voltages. Direct measurement

of the high or low voltages would include certain values of leakage electromotive force

(EMF). In order to discover the relationship between vibration changes and voltage

fluctuation, which is induced only by main flux variation in the transformer core, three

belt coils were utilised to exclude the effects of leakage flux.

Figure B4. Schematic for main flux variation measurement through belt coil–inducted EMF.

Three belt coils were installed in the left, central, and right limbs separately.

Each coil was wound clockwise with seven turns at the same location close to the top

yoke (see the image in Figure B2). The overall dimensions of the core and belt coil

locations are shown in Figure B4.

155

Belt coil–inducted EMFs were recorded simultaneously with high and low

voltages, currents, and accelerations. Although the EMF induced by the belt coils

excluded the influence of leakage flux, additional installations of such belt coils would

cause unavoidable modifications to the power transformer. In practice, voltage

measurement in the transformer’s HV/LV ends is more convenient. To avoid these

modifications, a test was designed to verify whether the load voltages could be

substitute parameters for measurement of flux variation.

Figure B5. Induced voltage of the centre belt coil and HV winding voltage under different load statuses (U1: 250 V, U2: 350 V, and U3: 450 V).

In this test, induced voltages in the centre belt coil and the HV winding were

measured under five loading statuses controlled by switching on different heater groups.

The five loading statuses from the open circuit to a 13.24-Ω resistive load are defined as

L1 to L5. Figure B5 shows the normalised voltages for the five load statuses at three

operating voltages. Each voltage recorded from both belt coils and the transformer

winding was normalised with the voltage of the open-circuit load status. The dashed

lines show induced voltages in the centre belt coil at input voltages of 250 V, 350 V,

and 450 V. In general, those induced voltages all decrease with the loading status from

0 0.02 0.04 0.06 0.080.97

0.975

0.98

0.985

0.99

0.995

1

Loading Status [-1]

Nor

mal

ized

Vol

tage

[%]

Bc U1Bc U2Bc U3HV U1HV U2HV U3

156

the open circuit to the full load. The solid lines show voltage variation in the HV

winding at the three input voltages. As the HV is induced from the centre belt coil, it

decreases with the loading status. The measured voltage difference between the belt coil

and the HV winding can be found in Figure B5. Since leakage flux is typically larger

when the ferromagnetic core becomes saturated, the voltage tendencies at the 250 V and

350 V operating voltages show smaller discrepancies than those of the 450 V operating

voltage. These differences mainly come from the effects of leakage flux.

The above discussion suggests that the primary voltage (i.e., HV) variations can

also be used as an estimator of the fluctuation of magnetic flux when direct

measurements in an in-service transformer are inconvenient.

B3.1.2 Power source instability and its relationship to transformer vibration

For an in-service transformer, there is a voltage variation of around 3%, as found

in Figure B1. To simulate the power source variation in the model transformer, a

variable transformer was installed to provide certain voltage fluctuations and flux

variation. Since flux density and its distribution within a transformer core determine the

core vibration, voltage fluctuation might be related to the transformer’s vibration status.

In this paper, the sum of all of the testing points’ squared accelerations, 8

2

1i

iA

, is

employed to describe transformer vibration. Figure B6 shows the voltage and vibration

variations at three operating voltages. The voltage variations are normalised by the

minimum voltages of each operating voltage.

157

Figure B6. Power source instability and corresponding effects on transformer vibration.

Power source instability in terms of voltage variation causes 5.8%, 3.59%, and

2.58% maximum differences at 250 V, 350 V, and 450 V operating voltages,

respectively. With regard to these voltage fluctuations, the changes in transformer

vibration are found to be 0.89 dB, 0.64 dB, and 0.84 dB. For all three operating voltages,

the vibration fluctuates in the same way as voltage variation.

For further investigation of the relationship between transformer voltage and

vibration, the correlation coefficients between them were calculated at three operating

2 4 6 8 10 120

1

2

3

4

5

6

Sampling Number

Vol

tage

Var

iatio

n [%

]

250V350V450V

2 4 6 8 10 12-50

-45

-40

-35

-30

-25

Sampling Number

acc.

2 [dB

]

250V350V450V

158

voltages. Table B1 shows the correlation coefficients between transformer vibration and

voltage fluctuation tendencies.

Table B1. Correlation coefficients between voltage and vibration fluctuations. Operating voltage 250 V 350 V 450 V

Correlation coefficient 0.974 0.9605 0.9148

Table B1 shows the high correlation coefficients between vibration fluctuation

and voltage variation due to power instability. Since the only parameter changed in each

test is voltage fluctuation, the vibration variation can be readily related to the changes in

the transformer voltage.

B3.1.3 Loading shift–induced voltage fluctuation and its relationship to transformer vibration

A comparison between the trend in voltage variation and the resulting vibration

differences can be seen in Figure B7. The voltage variations at the three operating

voltages were normalised with voltages in the L5 loading status (full loading) by:

5

5

100%, 1,2,3,4Li Li

L

V VV iV

, (B1)

where the reference voltages 5LV for the three operating voltages are 463.9 V, 371.7 V,

and 260.5 V. The voltage variations will be made larger when switching between large

loading spans. Figure B7 shows a clear decrease in primary voltages, while the

maximum voltage differences are 2.96%, 2.29%, and 2.1% as the resistive load

decreases at the fundamental frequency. The overall trend in the transformer vibration is

one of decline with the increase in resistive load. The maximum difference between the

open circuit and the full loading status is approximately 1 dB. As described in Section

159

3.1.1, a similar relationship can be found between the voltage variation and changes in

transformer vibration in Figure B7.

Figure B7. Loading shift–induced voltage variation and the corresponding changes in transformer vibration.

As the stability of the primary voltage supply can be reflected by the transformer

input voltage, the measurements of voltage and vibration at different loading statuses

were conducted within a short time period and repeated more than three times to verify

0 0.02 0.04 0.06 0.080

0.5

1

1.5

2

2.5

3

Loading Status [-1]

Vol

tage

Var

iatio

n [%

]

250V350V450V

0 0.02 0.04 0.06 0.08-45

-40

-35

-30

-25

Loading Status [-1]

acc.

2 [dB

]

250V350V450V

160

the real variations. In a short time period, a constant voltage supply to the transformer is

assumed.

A general understanding of voltage variation with loading status can be gained

by analysing the input impedance of the transformer circuit of an ideal transformer. To

the power source, the transformer acts like an effective loading with input impedance

iZ . A voltage drop across iZ can be described as an equivalent EMF, 0E , with respect

to the downstream loading. The input impedance can be calculated from the transformer

equivalent circuit [B15]. Only inductive and resistive loadings are considered in the

following equation for calculating the transformer’s input impedance, as they are

representative load types in commercial applications:

21 2

1 12 2R Ri

L L

L LZ R j Lj L j L

,

(B2)

where:

1R , 2R : Primary and secondary resistances;

1L , 2L : Primary and secondary inductances; and

LL , LR : Loading inductance and resistance.

As discussed above, a constant voltage supply, sE , can be assumed in a short time

period. Therefore, the equivalent EMF, 0E , determined by iZ can be affected by

loading status.

For this test, since 1R and 2R are negligible compared with the winding

inductive impedance, only resistive loads are connected in the LV end ( 0LL ). The

input impedance can be decomposed and simplified as:

2 21 2 1

2 2 2 2 2 22 2

RR R

Li L

L L

L L LZ R jL L

.

(B3)

161

The maximum measured resistive load, LR , is 75.7 Ω and is much smaller than the

secondary inductive impedance, 2L , 1068 Ω. Thus, Eq. (B3) can be rewritten as:

2 21 12 22 2

RLi L

N LZ R jN L

.

(B4)

It is clear that the input impedance is dependent on the resistive load. During the

experiment, parallel-connected resistive loads were gradually removed such that iR

increased. Therefore, the input impedance and correlated equivalent EMF would

increase accordingly.

In practice, voltage disturbances caused by customer demands are quite common.

Peak and valley electrical demands from each user will be different. Thus, any voltage

fluctuation would happen within a permitted range.

In the above cases, the vibration variation has nothing to do with structural

changes. It is caused by voltage fluctuations due to power instability or loading shift.

This fluctuation alters the magnetisation in the core and the current in the winding, and

then affects transformer vibration as well. However, vibration induced by core

magnetostriction and magnetostatic forces will be the first to be considered since

current-induced transformer vibration is typically small.

B3.2 Effects of structural changes on transformer vibration

Power transformers are often exposed to multiple short circuit shocks, insulation aging,

and repeated thermal processes. As a result, the mechanical strength of transformer

insulation becomes weak. Furthermore, mechanical defects and insulation weaknesses

affect each other. One of the consequences is a decrease in clamping pressures in both

the winding and core assembly. As part of the aim to study the variations in transformer

vibration due to structural changes, the reduction of winding and core clamping pressure

will be studied as a form of structural change in this section.

162

A vibration method based on vibration energy distribution and harmonics

analysis was described in Ref. [B13] for diagnosis of decreases in clamping force.

Coefficients of the winding and core clamping pressures were calculated per-unit using

thirty parameters, and specific criteria were selected to evaluate the degree of

transformer fixation. However, voltage fluctuation and its relation to vibration

characteristics were not been studied in detail.

Figure B8. Voltage and vibration fluctuations at different winding clamping forces.

0 20 40 60 80 1000

1

2

3

4

Winding Looseness [%]

Vol

tage

Var

iatio

n [%

]

250V350V450V

0 20 40 60 80 100-45

-40

-35

-30

-25

Winding Looseness [%]

acc.

2 [dB

]

250V350V450V

163

In this test, voltage fluctuation due to power grid instability was recorded while

the transformer vibrations were measured. Since the secondary loading was kept

constant (open circuit), voltage fluctuations mainly came from the power supply. Based

on the discussion in Section 3.1.2 about vibration changes due to power instability,

changes in transformer vibration induced by structural changes can be estimated.

As described in Figure B8, the normalised voltage fluctuates randomly under

different voltage ratings (normalised by the minimum voltages of each operating

voltage). On average, transformer vibration varies with its own tendencies when it is

affected by certain structural modifications. This modification is realised by reducing

the winding clamping force from 0% to 90% looseness.

Figure B8 shows that the voltage discrepancy between 10% and 20% winding

looseness is 1.88% at a 350-V operating voltage while the reduction in transformer

vibration is 3.78 dB. Referring to the voltage variation in Section 3.1.2, a 1.77% voltage

difference causes 0.52-dB changes in vibration from case 9 to case 10 at a 350-V

voltage input. Under the same voltage drop, the change in transformer vibration shows a

3.22 dB difference after introducing winding looseness. Compared to the vibration

difference induced by structural changes, the power instability–induced changes in

transformer vibration are relatively small. For the other winding clamping levels, further

variation in vibration can always be found by comparing with the power instability–

induced changes in transformer vibration. It is evident that the measured vibration

changes at different levels of winding looseness mainly come from structural changes

under small voltage fluctuations. The changes in the transformer’s mechanical

properties can be detected through vibration measurement. Therefore, monitoring the

transformer’s mechanical properties using vibration measurements becomes feasible.

164

For a mechanical failure that will produce large vibration differences, a diagnosis based

on the vibration method proposed in Ref. [B13] is capable of exact predictions.

As can be seen from Figure B8, transformer vibration does not always increase

as the winding clamping force is gradually reduced. A number of parameters are

changed as the clamping loosens: e.g., friction between turns, damping and stiffness of

winding stacks, and slight differences in dimensions caused by springback of insulation

materials. A study of the underlying causes that govern this process is still ongoing.

A statistical study showed that fewer transformer failures occur in the

transformer core than in the winding part [B16]. However, mechanical vibration is quite

sensitive to changes in core lamination, e.g., a joint design difference [B17]. Therefore,

any faults in the core would cause a significant impact on magnetic circuit parameters.

Since the flux density and its distribution greatly affect magnetostriction and

magnetostatic forces between the laminations, such changes in the magnetic circuit are

expected to produce certain variations in transformer vibration. In this part, the

clamping pressure applied to the core assembly is defined as a variable parameter to

simulate mechanical faults in the magnetic circuit. The voltage and vibration variations

are investigated together for a summary of their relationship.

165

Figure B9. Voltage and vibration fluctuations at different core clamping forces.

The voltage and vibration changes were measured for five levels of clamping

looseness between 0 to 100%. Power instability–induced voltage variation shows a

maximum 2% difference in Figure B8. The corresponding changes in transformer

vibration are within a 0.5-dB range based on the above conclusion. From 60% to 100%

clamping looseness, up to a 4.26 dB difference can be found in the transformer

vibration. However, vibration variations are only less than 1 dB for the first three core

clamping levels. In these cases, there are typically small contributions to the variation of

0 20 40 60 80 1000

0.5

1

1.5

2

Core Looseness [%]

Vol

tage

Var

iatio

n [%

]

250V350V450V

0 20 40 60 80 100-40

-35

-30

-25

-20

Core Looseness [%]

acc.

2 [dB

]

250V350V450V

166

transformer vibration from those structural changes. Therefore, a determination of such

mechanical failures becomes difficult.

Changes in transformer vibration induced by core clamping looseness can be

explained partly by domain theory and partly by changes in the mechanical properties.

The effects of stress on the magnetic properties of SiFe laminations have been studied

by several researchers [B18, B19]. Linear stress, normal stress, and a combination of

these applied to SiFe sheets all lead to a magnetoelastic energy increase in terms of

magnetostriction. Domains formed within the SiFe sheets orient themselves in

directions of easy magnetisation [B20]. An applied core clamping pressure destroys this

balance, the angles of easy directions deviate, and, hence, there is an increase in power

loss and magnetostriction. However, in practice, the sensitivity to normal stress would

be very low and only minor changes in the domain structure would be expected [B21].

A clamping force applied to the core assembly can only produce normal stresses.

There is no doubt that mechanical properties will change as the core clamping

pressure decreases from a fully tight level. As one source of core vibration, the

interlaminar magnetic force generates most of the out-of-plane vibration, which is

strongly affected by the core’s effective Young’s modulus, which in turn will be

reduced as a result of core looseness. In addition, the Young’s modulus of a single SiFe

sheet is also reduced with a small stress state.

As an indication of the magnetomechanical coupling efficiency, the

magnetomechanical coupling factor, k, is defined as:

2 1 H Bk Y Y , (B5)

where YH represents the Young’s modulus under the magnetically free condition, and YB

is that under the magnetically blocked condition. A corresponding experimental study

167

records that k will be lowered as a result of extra stress [B22], which indicates a lower

Young’s modulus with respect to the smaller clamping pressure. In this way, the energy

exchange between magnetic and mechanical energy will be more efficient and produce

greater core vibrations.

4. Conclusions

The above results lead to the following conclusions:

(1) The relative fluctuation of source voltage was around 3% and a maximum

fluctuation of 5.88% was measured. These voltage fluctuations induced a variation in

the transformer vibration of less than 0.89 dB.

(2) Loading switching generated a maximum voltage fluctuation of 2.96% and

induced vibration changes of less than 0.96 dB.

(3) When the system’s clamping force changed, the maximum change in voltage

was 3.25% and the maximum change in vibration was 4.16 dB, which was much higher

than the change induced by other fluctuations.

(4) As an indication of flux density, the induced voltage in the belt coil showed

good agreement with the transformer’s high and low voltages.

(5) A high correlation was found between voltage fluctuation and vibration

changes when there was no change in the mechanical and electrical parameters of the

transformer.

168

Reference

[B1] Boczar, T, Borucki, S, Cichon, A, & Lorenc, M 2008, ‘The analysis of mechanical

vibrations & acoustic pressure level of a transformer model’, Acta Physica Polonica A,

vol. 114, pp. 21–26.

[B2] Pan, J, Wang, J & Jin, M 2010, ‘Structural vibration of a transformer’,

Proceedings of the 17th international congress on sound & vibration, Cairo.

[B3] Weiser, B, Hasenzagl, A, Booth, T & Pfützner, H 1996, ‘Mechanisms of noise

generation of model transformer cores’, Journal of Magnetism & Magnetic Materials,

vol. 160, pp. 207–209.

[B4] Weiser, B, Pfützner, H & Anger, J 2000, ‘Relevance of magnetostriction and

forces for the generation of audible noise of transformer cores’, IEEE Transactions on

Magnetics, vol. 36, pp. 3759–3777.

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171

Nomenclature

Symbols

A magnetic vector potential

0A cross-sectional area

B magnetic field density

C closed curve

S surface

J current density

H magnetic field strength

magnetic permeability

0 space permeability

jm , kn location constant

1a , 1a LV winding radius

2a , 2a HV winding radius

t core window width

h , 1h , 2h core window height

dF force density

vector angles

n natural frequency

nL modal constant

E Young’s modulus

material density

I area moment of inertia

tilting angle, twist angle

d elongation & buckling

distances

MH mechanical FRF

EH electrical FRF

H vibration response function

ˆ ( )kF x force per unit volume at

location kx

V volume of the transformer

structure

( )oU primary voltage

o testing frequency

( )o magnetic flux

N number of winding turns

S cross-sectional area of the

core

M mechanical parameter vector

E electrical parameter vector

fn natural frequency shifts

172

n number of modes

ir i th component of mode

shape r

x t free vibration vector

mode shape matrix

r thr eigenvalue

N mode order

rd damped modal frequency

r modal damping ratio

rm rth modal mass

rn rth modal frequency

Abbreviations

2D Two-Dimensional

3D Three-Dimensional

CCF Cross-Correlation Function

DFE Double Fourier Expansion

DGA Dissolved Gas Analysis

D.O.F. Degrees of Freedom

EM ElectroMagnetic

EMA Experimental Modal Analysis

EMF Electromotive Force

FE Finite Element

FRF Frequency Response Function

FRA Frequency Response Analysis

FS Fluid-structural

HV High voltage

IRF Impulse Response Function

ITD Ibrahim Time Domain

kV kilovolt

kVA kilovolt-amperes

LV Low voltage

MS MagnetoStrictive

MVA Megavolt-amperes

MVP Magnetic Vector Potential

NExt Natural Excitation Technique

OMA Operational Modal Analysis

PSD Power Spectral Density

RVM Return Voltage Method

SiFe Silicon-iron

SS Structural-structural

TF Transfer Function

TCG Total Combustible Gases

WTHD Weighted Total Harmonic

Distortion

173

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184

Publications originated from this thesis

[1] Y. Wang, J. Pan and M. Jin, “Finite element modelling of the vibration of a power

transformer,” Proceedings of Acoustics 2011, Nov. 2011, Gold Coast, Australia.

[2] Y. Wang and J. Pan, “Voltage and vibration fluctuations in power transformers,”

Proceedings of Acoustics 2012, Nov. 2012, Fremantle, Australia.

[3] Y. Wang and J. Pan, “Applications of vibration modal parameter identification to

transformer condition monitoring,” Proceedings of the sixth world conference on SCM,

July, 2014, Barcelona, Spain.

[4] Y. Wang and J. Pan, “Comparison of Mechanically and Electrically Excited

Vibration Frequency Responses of a Small Distribution Transformer,” IEEE

Transactions on Power Delivery, 2015 (in press).

[5] Y. Wang and J. Pan, “Applications of Operational Modal Analysis to Transformer

Condition Monitoring,” IEEE Transactions on Power Delivery, 2015 (in press).

[6] J. Pan, J. Ming and Y. Wang, “Vibration of Power Transformers and its Application

for Condition Monitoring,” Proceedings of the 14th Asia Pacific Vibration Conference

Dec. 2011, Hong Kong.

[7] J. Pan, J. Ming and Y. Wang, “Electrical and Vibration Characteristics of Traction

Transformers,” Proceedings of the 20th International Congress on Sound and Vibration,

July 2013, Bangkok, Thailand.

transformer vibration and its application to condition monitoring

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